metric aspects in algebraic geometry, on the average ... · (celebrating the work of mike shub)...
TRANSCRIPT
![Page 1: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/1.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
“Metric Aspects in Algebraic Geometry,on the Average”.
(Celebrating the work of Mike Shub)
Luis M. Pardo1
May, 2012
1Univ. de Cantabria.Luis M. Pardo MAAGA, Mike’s May 68
![Page 2: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/2.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Mike’s Influence
Mike’s ideas have strongly influenced my work in the last decade.
Specially, his work with Steve Smale on Numerical Solving ofPolynomial Equations.
His influence and their work was essential to deal with Smale’s 17thProblem : [Beltran-P., 2009]...
But these are already “old” mathematical results. I wanted somethingfresh, specifically oriented for this conference in Mike’s honor.
Thus, I tried to work on some (maybe modest and preliminary)results based on ideas from Mike’s work.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 3: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/3.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Mike’s Influence
Mike’s ideas have strongly influenced my work in the last decade.
Specially, his work with Steve Smale on Numerical Solving ofPolynomial Equations.
His influence and their work was essential to deal with Smale’s 17thProblem : [Beltran-P., 2009]...
But these are already “old” mathematical results. I wanted somethingfresh, specifically oriented for this conference in Mike’s honor.
Thus, I tried to work on some (maybe modest and preliminary)results based on ideas from Mike’s work.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 4: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/4.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Mike’s Influence
Mike’s ideas have strongly influenced my work in the last decade.
Specially, his work with Steve Smale on Numerical Solving ofPolynomial Equations.
His influence and their work was essential to deal with Smale’s 17thProblem : [Beltran-P., 2009]...
But these are already “old” mathematical results. I wanted somethingfresh, specifically oriented for this conference in Mike’s honor.
Thus, I tried to work on some (maybe modest and preliminary)results based on ideas from Mike’s work.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 5: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/5.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Mike’s Influence
Mike’s ideas have strongly influenced my work in the last decade.
Specially, his work with Steve Smale on Numerical Solving ofPolynomial Equations.
His influence and their work was essential to deal with Smale’s 17thProblem : [Beltran-P., 2009]...
But these are already “old” mathematical results.
I wanted somethingfresh, specifically oriented for this conference in Mike’s honor.
Thus, I tried to work on some (maybe modest and preliminary)results based on ideas from Mike’s work.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 6: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/6.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Mike’s Influence
Mike’s ideas have strongly influenced my work in the last decade.
Specially, his work with Steve Smale on Numerical Solving ofPolynomial Equations.
His influence and their work was essential to deal with Smale’s 17thProblem : [Beltran-P., 2009]...
But these are already “old” mathematical results. I wanted somethingfresh, specifically oriented for this conference in Mike’s honor.
Thus, I tried to work on some (maybe modest and preliminary)results based on ideas from Mike’s work.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 7: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/7.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Mike’s Influence
Mike’s ideas have strongly influenced my work in the last decade.
Specially, his work with Steve Smale on Numerical Solving ofPolynomial Equations.
His influence and their work was essential to deal with Smale’s 17thProblem : [Beltran-P., 2009]...
But these are already “old” mathematical results. I wanted somethingfresh, specifically oriented for this conference in Mike’s honor.
Thus, I tried to work on some (maybe modest and preliminary)results based on ideas from Mike’s work.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 8: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/8.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some Self-Constraints
* No Condition Number.
* No Complexity.
* No Homotopy/Path Continuation Methods.
* No Polynomial System Solving.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 9: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/9.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some Self-Constraints
* No Condition Number.
* No Complexity.
* No Homotopy/Path Continuation Methods.
* No Polynomial System Solving.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 10: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/10.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some Self-Constraints
* No Condition Number.
* No Complexity.
* No Homotopy/Path Continuation Methods.
* No Polynomial System Solving.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 11: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/11.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some Self-Constraints
* No Condition Number.
* No Complexity.
* No Homotopy/Path Continuation Methods.
* No Polynomial System Solving.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 12: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/12.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some Self-Constraints
* No Condition Number.
* No Complexity.
* No Homotopy/Path Continuation Methods.
* No Polynomial System Solving.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 13: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/13.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Mike’s Inspiring Source
L. Blum, M. Shub, Evaluating Rational Functions: Infinite Precisionis finite cost and Tractable on average, SIAM J. on Comput. 15(1986) 384–398.
Main outcome of this manuscript:
Theorem
volx ∈ B(0, r) : |Q(x)| < εvol[B(0, r)]
≤ CQε1/d
r,
where Q ∈ R[X1, . . . , Xn] is a polynomial of degree at most d andB(0, r) is the ball of radius r centered at the origin.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 14: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/14.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Mike’s Inspiring Source
L. Blum, M. Shub, Evaluating Rational Functions: Infinite Precisionis finite cost and Tractable on average, SIAM J. on Comput. 15(1986) 384–398.
Main outcome of this manuscript:
Theorem
volx ∈ B(0, r) : |Q(x)| < εvol[B(0, r)]
≤ CQε1/d
r,
where Q ∈ R[X1, . . . , Xn] is a polynomial of degree at most d andB(0, r) is the ball of radius r centered at the origin.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 15: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/15.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Mike’s Inspiring Source
L. Blum, M. Shub, Evaluating Rational Functions: Infinite Precisionis finite cost and Tractable on average, SIAM J. on Comput. 15(1986) 384–398.
Main outcome of this manuscript:
Theorem
volx ∈ B(0, r) : |Q(x)| < εvol[B(0, r)]
≤ CQε1/d
r,
where Q ∈ R[X1, . . . , Xn] is a polynomial of degree at most d andB(0, r) is the ball of radius r centered at the origin.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 16: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/16.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Transforming this outcome into a recipe for this conference
Put something concerning the growth of the absolute value ofmultivariate polynomials.
Add some average and probability.
And, finally, add some algebraic varieties and metrics and see whathappens...
Luis M. Pardo MAAGA, Mike’s May 68
![Page 17: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/17.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Transforming this outcome into a recipe for this conference
Put something concerning the growth of the absolute value ofmultivariate polynomials.
Add some average and probability.
And, finally, add some algebraic varieties and metrics and see whathappens...
Luis M. Pardo MAAGA, Mike’s May 68
![Page 18: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/18.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Transforming this outcome into a recipe for this conference
Put something concerning the growth of the absolute value ofmultivariate polynomials.
Add some average and probability.
And, finally, add some algebraic varieties and metrics and see whathappens...
Luis M. Pardo MAAGA, Mike’s May 68
![Page 19: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/19.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Main Topics of the Talk
On the Expected Growth of Multivariate Polynomials
On the Expected Separations of zeros of a polynomial system(illustrating Mike’s double fibration technique).
On the Expected Distance between two complex projectivevarieties (same technique).
On the Expected average height of resultants: An ArithmeticPoisson Formula for the Multi-variate Resultant.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 20: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/20.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Main Topics of the Talk
On the Expected Growth of Multivariate Polynomials
On the Expected Separations of zeros of a polynomial system(illustrating Mike’s double fibration technique).
On the Expected Distance between two complex projectivevarieties (same technique).
On the Expected average height of resultants: An ArithmeticPoisson Formula for the Multi-variate Resultant.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 21: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/21.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Main Topics of the Talk
On the Expected Growth of Multivariate Polynomials
On the Expected Separations of zeros of a polynomial system(illustrating Mike’s double fibration technique).
On the Expected Distance between two complex projectivevarieties (same technique).
On the Expected average height of resultants: An ArithmeticPoisson Formula for the Multi-variate Resultant.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 22: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/22.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Main Topics of the Talk
On the Expected Growth of Multivariate Polynomials
On the Expected Separations of zeros of a polynomial system(illustrating Mike’s double fibration technique).
On the Expected Distance between two complex projectivevarieties (same technique).
On the Expected average height of resultants:
An ArithmeticPoisson Formula for the Multi-variate Resultant.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 23: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/23.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Main Topics of the Talk
On the Expected Growth of Multivariate Polynomials
On the Expected Separations of zeros of a polynomial system(illustrating Mike’s double fibration technique).
On the Expected Distance between two complex projectivevarieties (same technique).
On the Expected average height of resultants: An ArithmeticPoisson Formula for the Multi-variate Resultant.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 24: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/24.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Basic Notations (I)
*(d):= (d1, . . . , dm) a list of degrees.
* X0, . . . , Xn:= A list of variables
* H(m)(d) := Lists (f1, . . . , fm) of m complex homogeneous polynomials
of respective degrees deg(fi) = di.
*P(m)(d) := Affine polynomials in X1, . . . , Xn with deg(fi) ≤ di.
*D(d):=∏mi=1 di the Bezout number.
* N := the complex dimension of H(m)(d) is N + 1.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 25: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/25.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Basic Notations (II)
*Pn(C):= P(Cn+1), the projective complex space,
*dR(x, y):= the Riemannian distance between two pointsx, y ∈ P(Cn+1) and
dP(x, y) := sin dR(x, y),
the “projective” distance.
*VP(f):= for f ∈ H(m)(d) , the projective variety (in Pn(C)) of the
common zeros of polynomials in the list f .
*VA(f):= for f ∈ P(m)(d) , the affine variety (in Cn) of the common zeros
of polynomials in the list f .
Luis M. Pardo MAAGA, Mike’s May 68
![Page 26: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/26.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Bombieri-Weil norm
An Hermitian form which is also an expectation:
||f ||2∆ :=
(d+ n
n
)1
vol[S2n+1]
∫S2n+1
|f(z)|2dνS(z) =
(d+ n
n
)ES2n+1 [|f |2].
* S(H(m)(d) ) := the sphere of radius one in H(m)
(d) with respect to
Bombieri’s norm || · ||∆.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 27: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/27.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Bombieri-Weil norm
An Hermitian form which is also an expectation:
||f ||2∆ :=
(d+ n
n
)1
vol[S2n+1]
∫S2n+1
|f(z)|2dνS(z) =
(d+ n
n
)ES2n+1 [|f |2].
* S(H(m)(d) ) := the sphere of radius one in H(m)
(d) with respect to
Bombieri’s norm || · ||∆.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 28: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/28.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Growth of Polynomials
Expected Growth of Polynomials
Luis M. Pardo MAAGA, Mike’s May 68
![Page 29: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/29.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
An almost immediate statement
An Estimate
Assume deg(f) = d ∈ 2Z is of even degree. Then,
Eγ [|f(z)|] ≤ ||f ||∆
d/2∑k=0
(d/2
k
)2k−n−1Γ(n+ k + 1)
πnΓ(n+ 1)
,
where Eγ is the expectation with respect to the Gaussiandistribution in Cn.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 30: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/30.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Let’s look for something a little bit sharper (I)
Assume Cn is endowed with the pull–back distribution induced by thecanonical embedding ϕ0 : Cn −→ Pn(C).
Change |f(z)| by log |f(z)|,and change the role of ||f ||∆. Then, study:
E := Ef∈S(P
(n)d )
[ECn [log |f |]],
where S(P(n)d ) is the sphere of radius one with respect to
Bombieri–Weil norm.
This is the average value of something similar to the marking time in[Blum-Shub, 86].
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Let’s look for something a little bit sharper (I)
Assume Cn is endowed with the pull–back distribution induced by thecanonical embedding ϕ0 : Cn −→ Pn(C). Change |f(z)| by log |f(z)|,
and change the role of ||f ||∆. Then, study:
E := Ef∈S(P
(n)d )
[ECn [log |f |]],
where S(P(n)d ) is the sphere of radius one with respect to
Bombieri–Weil norm.
This is the average value of something similar to the marking time in[Blum-Shub, 86].
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Let’s look for something a little bit sharper (I)
Assume Cn is endowed with the pull–back distribution induced by thecanonical embedding ϕ0 : Cn −→ Pn(C). Change |f(z)| by log |f(z)|,and change the role of ||f ||∆.
Then, study:
E := Ef∈S(P
(n)d )
[ECn [log |f |]],
where S(P(n)d ) is the sphere of radius one with respect to
Bombieri–Weil norm.
This is the average value of something similar to the marking time in[Blum-Shub, 86].
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Let’s look for something a little bit sharper (I)
Assume Cn is endowed with the pull–back distribution induced by thecanonical embedding ϕ0 : Cn −→ Pn(C). Change |f(z)| by log |f(z)|,and change the role of ||f ||∆. Then, study:
E := Ef∈S(P
(n)d )
[ECn [log |f |]],
where S(P(n)d ) is the sphere of radius one with respect to
Bombieri–Weil norm.
This is the average value of something similar to the marking time in[Blum-Shub, 86].
Luis M. Pardo MAAGA, Mike’s May 68
![Page 34: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/34.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Let’s look for something a little bit sharper (I)
Assume Cn is endowed with the pull–back distribution induced by thecanonical embedding ϕ0 : Cn −→ Pn(C). Change |f(z)| by log |f(z)|,and change the role of ||f ||∆. Then, study:
E := Ef∈S(P
(n)d )
[ECn [log |f |]],
where S(P(n)d ) is the sphere of radius one with respect to
Bombieri–Weil norm.
This is the average value of something similar to the marking time in[Blum-Shub, 86].
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
A little bit sharper (II)
Proposition
With these notations we have:
E :=1
2(dHn −HR) ,
where Hr is the r−th harmonic number and R :=(d+nn
)
Recall that
Hr ≈ log(r) + γ +O(1
r),
where γ is Euler-Mascheroni number and, hence
E ≈ d
2log
(dn
d+ n
).
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
A little bit sharper (II)
Proposition
With these notations we have:
E :=1
2(dHn −HR) ,
where Hr is the r−th harmonic number and R :=(d+nn
)Recall that
Hr ≈ log(r) + γ +O(1
r),
where γ is Euler-Mascheroni number and, hence
E ≈ d
2log
(dn
d+ n
).
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
An example: A Complementary of [Blum-Shub, 86]
Corollary
E := Ef∈S(P
(n)d )
[ECn [|f |−1]] ≥ e− d2 (Hn−HR/d),
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Other “potential” applications (not finished yet)
Motivated by the talk by Diego Armentano and question (d) in[Armentano-Shub, 12]...slighty modified...
Hints on I(f), for
f ∈ H(n)(d) , and:
I(f) :=
∫S2n+1
e||f(z)||2
2
||f(z)||2n−1dz.
A good “hint” could be the Expectation ES(H(d))[I(f)] and this isgiven by the following:
ES(Hd)[I(f)] =
∞∑k=0
ν2n+1
k!2k
∞∑j=0
(2k − n+ 1)j
j!ES(H(d))[ES2n+1 [logj ||f ||]]
.
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Other “potential” applications (not finished yet)
Motivated by the talk by Diego Armentano and question (d) in[Armentano-Shub, 12]...slighty modified... Hints on I(f), for
f ∈ H(n)(d) , and:
I(f) :=
∫S2n+1
e||f(z)||2
2
||f(z)||2n−1dz.
A good “hint” could be the Expectation ES(H(d))[I(f)] and this isgiven by the following:
ES(Hd)[I(f)] =
∞∑k=0
ν2n+1
k!2k
∞∑j=0
(2k − n+ 1)j
j!ES(H(d))[ES2n+1 [logj ||f ||]]
.
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Other “potential” applications (not finished yet)
Motivated by the talk by Diego Armentano and question (d) in[Armentano-Shub, 12]...slighty modified... Hints on I(f), for
f ∈ H(n)(d) , and:
I(f) :=
∫S2n+1
e||f(z)||2
2
||f(z)||2n−1dz.
A good “hint” could be the Expectation ES(H(d))[I(f)] and this isgiven by the following:
ES(Hd)[I(f)] =
∞∑k=0
ν2n+1
k!2k
∞∑j=0
(2k − n+ 1)j
j!ES(H(d))[ES2n+1 [logj ||f ||]]
.
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Other “potential” applications (not finished yet)
Motivated by the talk by Diego Armentano and question (d) in[Armentano-Shub, 12]...slighty modified... Hints on I(f), for
f ∈ H(n)(d) , and:
I(f) :=
∫S2n+1
e||f(z)||2
2
||f(z)||2n−1dz.
A good “hint” could be the Expectation ES(H(d))[I(f)] and this isgiven by the following:
ES(Hd)[I(f)] =
∞∑k=0
ν2n+1
k!2k
∞∑j=0
(2k − n+ 1)j
j!ES(H(d))[ES2n+1 [logj ||f ||]]
.
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Separation of Solutions
On the average separation of the solutions
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
A Technique: solution variaties like as desing. a la Room-Kempf (I)
Double fibration (introduced and used by M. Shub and S. Smale intheir Bezout series) with deep and interesting consequences, whencombined with Federer’s co-area formula.
We consider the (smooth) solution variety:
V(m)(d) = (f, ζ) ∈ P(H(m)
(d) )× Pn(C) : ζ ∈ VP(f) ⊆ P(H(m)(d) )× Pn(C).
and we consider the two canonical projections:
V(m)(d)
π1 π2
P(H(m)(d) ) Pn(C)
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
A Technique: solution variaties like as desing. a la Room-Kempf (I)
Double fibration (introduced and used by M. Shub and S. Smale intheir Bezout series) with deep and interesting consequences, whencombined with Federer’s co-area formula.
We consider the (smooth) solution variety:
V(m)(d) = (f, ζ) ∈ P(H(m)
(d) )× Pn(C) : ζ ∈ VP(f) ⊆ P(H(m)(d) )× Pn(C).
and we consider the two canonical projections:
V(m)(d)
π1 π2
P(H(m)(d) ) Pn(C)
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
A Technique: solution variaties like as desing. a la Room-Kempf (I)
Double fibration (introduced and used by M. Shub and S. Smale intheir Bezout series) with deep and interesting consequences, whencombined with Federer’s co-area formula.
We consider the (smooth) solution variety:
V(m)(d) = (f, ζ) ∈ P(H(m)
(d) )× Pn(C) : ζ ∈ VP(f) ⊆ P(H(m)(d) )× Pn(C).
and we consider the two canonical projections:
V(m)(d)
π1 π2
P(H(m)(d) ) Pn(C)
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
A Technique: solution variaties like as desing. a la Room-Kempf (II)
* π−12 (x) := is a “linear” (of co–dimension m) in P(H(m)
(d) ).
* π−11 (f) := is the set of common zeros VP(f1, . . . , fm) and it is
“generically” a smooth projective variety of co–dimension m.
* For m ≤ n, π1 is onto.
Shub–Smale’s idea
Averaging in P(H(m)(d) ) may be translated to averaging in Pn(C)
through this double fibration.
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
A Technique: solution variaties like as desing. a la Room-Kempf (II)
* π−12 (x) := is a “linear” (of co–dimension m) in P(H(m)
(d) ).
* π−11 (f) := is the set of common zeros VP(f1, . . . , fm) and it is
“generically” a smooth projective variety of co–dimension m.
* For m ≤ n, π1 is onto.
Shub–Smale’s idea
Averaging in P(H(m)(d) ) may be translated to averaging in Pn(C)
through this double fibration.
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Separation of Solutions
For a zero–dimensional complete intersection variety VP(f) ⊆ Pn(C),the separation among its zeros:
sep(f) := mindP(ζ, ζ ′) : ζ, ζ ′ ∈ VP(f), ζ 6= ζ ′.
Lower bounds for these quantity are due to many authors:
The Davenport–Mahler-Mignotte lower bound for the univariatecase: Ω(2−d
2
)
Other authors (Dedieu, Emiris, Mourrain, Tsigaridas, ...) havealso treated the multivariate 2case:
Ω(2−D(d)).
2In [Castro–Haegele–Morais, p., 01] we also exhibited examples where this lowerbounds is achieved
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Separation of Solutions: An algorithmic question
* f ∈ H(d)
* z1, z2 ∈ Pn(C) that satisfy α−Theorem ([Shub–Smale]):
α(f, z1) ≤ α0, α(f, z2) ≤ α0.
Decide whether:
limk 7→∞
Nf (z1) = limk 7→∞
Nf (z2)?.
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Separation of Solutions: An algorithm
* t ∈ Neval
z(t)1 := N t
f (z1), z(t)2 := N t
f (z2).
if dP(z(t)1 , z
(t)2 ) > 2
22t−1 , then Output: They approach DIFFERENTzeros of f
else, Output: They approach the same zero of ffi
The algorithm works provided that:
sep(f) >4
22t−1 .
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Separation of Solutions: An algorithm
* t ∈ Neval
z(t)1 := N t
f (z1), z(t)2 := N t
f (z2).
if dP(z(t)1 , z
(t)2 ) > 2
22t−1 , then Output: They approach DIFFERENTzeros of f
else, Output: They approach the same zero of ffi
The algorithm works provided that:
sep(f) >4
22t−1 .
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Expected Minimum Separation (I)
For a zero–dimensional complete Intersection variety VP(f) ⊆ Pn(C),the “average” separation:
sepav(f) :=1
D(d)(D(d) − 1)
∑ζ,ζ′∈VP(f),ζ 6=ζ′
dP(ζ, ζ ′).
Theorem
Then, the following inequality holds:
EP(H(d))[sepav] ≥ 1
2
√1
d3(N + 1/2)n.
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Expected Minimum Separation (II)
What about the minimum separation of solutions?
sepmin(f) := minζ,ζ′∈VP(f),ζ 6=ζ′
dP(ζ, ζ ′).
Theorem
The following inequality holds:
ES(H(d)) [sepmin(f)] ≥ 1
4eD(d)d3/2(N + 1/2)−1/2.
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Expected Minimum Separation (II)
What about the minimum separation of solutions?
sepmin(f) := minζ,ζ′∈VP(f),ζ 6=ζ′
dP(ζ, ζ ′).
Theorem
The following inequality holds:
ES(H(d)) [sepmin(f)] ≥ 1
4eD(d)d3/2(N + 1/2)−1/2.
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Distance between two Complete Intersections
On the expected distance between two CompleteIntersections
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Expected Distance between two Complete Intersections
Let us consider two projective complete intersection varieties:
VP(f) := z ∈ Pn(C) : fi(z) = 0, 1 ≤ i ≤ m.
VP(g) := z ∈ Pn(C) : gj(z) = 0, 1 ≤ j ≤ s.
Average distance between VP(f) and VP(g) as
Dav(VP(f), VP(g)) :=1
vol[VP(f)]vol[VP(g)]
∫VP(f)×VP(g)
dP(x, y)dVP(f)dVP(g).
Theorem
With these notations, we have:
Ef,g[Dav(VP(f), VP(g))] = (1− 1
n+ 2).
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Expected Distance between two Complete Intersections
Let us consider two projective complete intersection varieties:
VP(f) := z ∈ Pn(C) : fi(z) = 0, 1 ≤ i ≤ m.
VP(g) := z ∈ Pn(C) : gj(z) = 0, 1 ≤ j ≤ s.
Average distance between VP(f) and VP(g) as
Dav(VP(f), VP(g)) :=1
vol[VP(f)]vol[VP(g)]
∫VP(f)×VP(g)
dP(x, y)dVP(f)dVP(g).
Theorem
With these notations, we have:
Ef,g[Dav(VP(f), VP(g))] = (1− 1
n+ 2).
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Expected Distance between two Complete Intersections
Let us consider two projective complete intersection varieties:
VP(f) := z ∈ Pn(C) : fi(z) = 0, 1 ≤ i ≤ m.
VP(g) := z ∈ Pn(C) : gj(z) = 0, 1 ≤ j ≤ s.
Average distance between VP(f) and VP(g) as
Dav(VP(f), VP(g)) :=1
vol[VP(f)]vol[VP(g)]
∫VP(f)×VP(g)
dP(x, y)dVP(f)dVP(g).
Theorem
With these notations, we have:
Ef,g[Dav(VP(f), VP(g))] = (1− 1
n+ 2).
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Expected Distance between two Complete Intersections (II)
Same notations. Assume VP(f) is zero–dimensional and s ≥ 1 (i.e.V P(f) ∩ VP(g) = ∅ a.e.).Distance between VP(f) and VP(g) as
dP(VP(f), VP(g)) := mindP(x, y) : x ∈ VP(f), y ∈ VP(g).
Theorem
Assume VP(g) is of co–dimension s and deg(VP(g)) := D′ =∏si=1 d
′i,
we have
Ef,g[dP(VP(f), VP(g))] ≥ 2s− 1
D(d)
(1 + 2
(∏si=1
d′ie2
s2
)) ,Moreover, for s ≥ 3, this may be rewritten as
Ef,g[dP(VP(f), VP(g))] ≥ 2s− 1
D(d) + 2deg(VP(g))D(d).
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Expected Distance between two Complete Intersections (II)
Same notations. Assume VP(f) is zero–dimensional and s ≥ 1 (i.e.V P(f) ∩ VP(g) = ∅ a.e.).Distance between VP(f) and VP(g) as
dP(VP(f), VP(g)) := mindP(x, y) : x ∈ VP(f), y ∈ VP(g).
Theorem
Assume VP(g) is of co–dimension s and deg(VP(g)) := D′ =∏si=1 d
′i,
we have
Ef,g[dP(VP(f), VP(g))] ≥ 2s− 1
D(d)
(1 + 2
(∏si=1
d′ie2
s2
)) ,Moreover, for s ≥ 3, this may be rewritten as
Ef,g[dP(VP(f), VP(g))] ≥ 2s− 1
D(d) + 2deg(VP(g))D(d).
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
On the Height of the Multi-variate Resultant Variety
On the Height of the Multi-variate Resultant: an ArithmeticPoisson formula
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Multi-variate Resultant (I)
We recall the “solution variety” in the case of over–determinedsystems:
V(n+1)(d) = (f, ζ) ∈ P(H(n+1)
(d) )×Pn(C) : ζ ∈ VP(f) ⊆ P(H(n+1)(d) )×Pn(C).
and we also consider the two canonical projections:
V(n+1)(d)
π1 π2
P(H(n+1)(d) ) Pn(C)
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Multi-variate Resultant (II)
* π−12 (x) := is a “linear” (of co–dimension n+ 1) in P(H(n+1)
(d) ).
* π−11 (f) := is the set of common zeros VP(f0, . . . , fn) and it is either
∅ or “generically” a single point.
* π1(V(n+1)(d) ) := R
(n+1)(d) is an irreducible complex hyper–surface,
usually known as the multi–variate resultant variety.
* Multi–variate resultant Res(n+1)(d) := is the multi–homogeneous
irreducible polynomial which defines R(n+1)(d) .
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Multi-variate Resultant (III)
Multi-variate Resultant and Resultant Variety R(n+1)(d) is a classical
object in Elimination Theory (also Computational AlgebraicGeometry, MEGA...).
It has been studied by many authors since XIX-th century withdifferent approaches and variations: Bezout, Sylvester, Macaulay,Chow,..., and, more recently, Jouanolou, Chardin, Gelfand, Kapranov,Zelevinsky, Sturmfels, Rojas, Heintz, Giusti, Dickenstein, D’Andrea,Krick, Szanto, Sombra...
The list is too long to be complete...
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Multi-variate Resultant (III)
Multi-variate Resultant and Resultant Variety R(n+1)(d) is a classical
object in Elimination Theory (also Computational AlgebraicGeometry, MEGA...).
It has been studied by many authors since XIX-th century withdifferent approaches and variations: Bezout, Sylvester, Macaulay,Chow,..., and, more recently, Jouanolou, Chardin, Gelfand, Kapranov,Zelevinsky, Sturmfels, Rojas, Heintz, Giusti, Dickenstein, D’Andrea,Krick, Szanto, Sombra...
The list is too long to be complete...
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Height of the Multi-Variate Resultant (I)
* Height of the multivariate resultant is an attempt to measure thelength of the integer coefficients...
It has interesting applications both in complexity and arithmeticgeometry.
With several variations (Chow forms, elimination polynomials,Arithmetic Nullstellensatz...) it has been studied by many authors:Nesterenko, Philippon, Krick, Sombra, D’Andrea, Remond, P. ...
Multi–variate resultants satisfy a Poisson formula which is a helpfulstatement for the knowledge of its properties.
Here, we are modest: we focus on the arithmetic version of PoissonFormula, whose geometric (degree) property can be stated as follows:
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Height of the Multi-Variate Resultant (I)
* Height of the multivariate resultant is an attempt to measure thelength of the integer coefficients...
It has interesting applications both in complexity and arithmeticgeometry.
With several variations (Chow forms, elimination polynomials,Arithmetic Nullstellensatz...) it has been studied by many authors:Nesterenko, Philippon, Krick, Sombra, D’Andrea, Remond, P. ...
Multi–variate resultants satisfy a Poisson formula which is a helpfulstatement for the knowledge of its properties.
Here, we are modest: we focus on the arithmetic version of PoissonFormula, whose geometric (degree) property can be stated as follows:
Luis M. Pardo MAAGA, Mike’s May 68
![Page 68: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/68.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Height of the Multi-Variate Resultant (I)
* Height of the multivariate resultant is an attempt to measure thelength of the integer coefficients...
It has interesting applications both in complexity and arithmeticgeometry.
With several variations (Chow forms, elimination polynomials,Arithmetic Nullstellensatz...) it has been studied by many authors:Nesterenko, Philippon, Krick, Sombra, D’Andrea, Remond, P. ...
Multi–variate resultants satisfy a Poisson formula which is a helpfulstatement for the knowledge of its properties.
Here, we are modest: we focus on the arithmetic version of PoissonFormula, whose geometric (degree) property can be stated as follows:
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
“Geometric” Poisson Formula
Let (d) := (d0, d1, . . . , dn) be a degree list, and (d′) := (d1, . . . , dn).
Let Res(n+1)(d) be the resultant associated to (d) in n+ 1 variables and
Res(n)(d′) the corresponding one associated to (d′).
For instance, Poisson’s Formula implies:
“Geometric” Poisson Formula
With these notations we have:
deg(Res(n+1)(d) ) = d0deg(Res
(n)(d′)) +D(d′),
where D(d′) :=∏ni=1 di is the Bezout number.
Inductively, we conclude :
deg(Res(n+1)(d) ) =
n∑i=0
∏j 6=i
dj .
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
“Geometric” Poisson Formula
Let (d) := (d0, d1, . . . , dn) be a degree list, and (d′) := (d1, . . . , dn).
Let Res(n+1)(d) be the resultant associated to (d) in n+ 1 variables and
Res(n)(d′) the corresponding one associated to (d′).
For instance, Poisson’s Formula implies:
“Geometric” Poisson Formula
With these notations we have:
deg(Res(n+1)(d) ) = d0deg(Res
(n)(d′)) +D(d′),
where D(d′) :=∏ni=1 di is the Bezout number.
Inductively, we conclude :
deg(Res(n+1)(d) ) =
n∑i=0
∏j 6=i
dj .
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
“Geometric” Poisson Formula
Let (d) := (d0, d1, . . . , dn) be a degree list, and (d′) := (d1, . . . , dn).
Let Res(n+1)(d) be the resultant associated to (d) in n+ 1 variables and
Res(n)(d′) the corresponding one associated to (d′).
For instance, Poisson’s Formula implies:
“Geometric” Poisson Formula
With these notations we have:
deg(Res(n+1)(d) ) = d0deg(Res
(n)(d′)) +D(d′),
where D(d′) :=∏ni=1 di is the Bezout number.
Inductively, we conclude :
deg(Res(n+1)(d) ) =
n∑i=0
∏j 6=i
dj .
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
“Geometric” Poisson Formula
Let (d) := (d0, d1, . . . , dn) be a degree list, and (d′) := (d1, . . . , dn).
Let Res(n+1)(d) be the resultant associated to (d) in n+ 1 variables and
Res(n)(d′) the corresponding one associated to (d′).
For instance, Poisson’s Formula implies:
“Geometric” Poisson Formula
With these notations we have:
deg(Res(n+1)(d) ) = d0deg(Res
(n)(d′)) +D(d′),
where D(d′) :=∏ni=1 di is the Bezout number.
Inductively, we conclude :
deg(Res(n+1)(d) ) =
n∑i=0
∏j 6=i
dj .
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Height of the Multi-Variate Resultant (I)
We may define the logarithmic height of the multi–resultant varietyeither following any of the usual definitions [Philippon, 91], [Bost,Gillet, Soule, 94], [Remon,01], [McKinnon, 01], [D’Andrea, Krick,Sombra, 11]....We just modify them by using the unitarily invariant height
htu(R(n+1)(d) ):
The only difference with “usual” notions is that we take into accountBombieri’s metric in the logarithmic Mahler’s measure (instead of theusual Hermitian product).
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Height of the Multi-Variate Resultant (I)
We may define the logarithmic height of the multi–resultant varietyeither following any of the usual definitions [Philippon, 91], [Bost,Gillet, Soule, 94], [Remon,01], [McKinnon, 01], [D’Andrea, Krick,Sombra, 11]....We just modify them by using the unitarily invariant height
htu(R(n+1)(d) ):
The only difference with “usual” notions is that we take into accountBombieri’s metric in the logarithmic Mahler’s measure (instead of theusual Hermitian product).
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
(Unitarily invariant) Logarithmic Mahler measure
Define the (unitarily invariant) logarithmic Mahler measure :
mS
(n+1)
(d)
(Res(n+1)(d) ) := E
S(n+1)
(d)
[log |Res(n+1)
(d) (f0, . . . , fn)|]
where
S(n+1)(d) :=
n∏i=0
S(Hdi).
For technical reasons, define:
R(n+1)(d) :=
htu(R(n+1)(d) )
D(d),
Note that this quantity only depends of (d) := (d0, . . . , dn) the“degree” list.
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
(Unitarily invariant) Logarithmic Mahler measure
Define the (unitarily invariant) logarithmic Mahler measure :
mS
(n+1)
(d)
(Res(n+1)(d) ) := E
S(n+1)
(d)
[log |Res(n+1)
(d) (f0, . . . , fn)|]
where
S(n+1)(d) :=
n∏i=0
S(Hdi).
For technical reasons, define:
R(n+1)(d) :=
htu(R(n+1)(d) )
D(d),
Note that this quantity only depends of (d) := (d0, . . . , dn) the“degree” list.
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Arithmetic Poisson Formula
Theorem
With these notations we have:
R(n+1)(d) =
n∏i=1
(n
di + n
)(R
(n)(d′) + I(d)
)+
E
d0,
where (d′) := (d1, . . . , dn), and
E
d0=
1
2(Hn −
HR
d0)
−1
4≤ I(d) ≤ 0.
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Arithmetic Poisson Formula
Theorem
With these notations we have:
R(n+1)(d) =
n∏i=1
(n
di + n
)(R
(n)(d′) + I(d)
)+
E
d0,
where (d′) := (d1, . . . , dn), and
E
d0=
1
2(Hn −
HR
d0)
−1
4≤ I(d) ≤ 0.
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Arithmetic Poisson Formula
Theorem
With these notations we have:
R(n+1)(d) =
n∏i=1
(n
di + n
)(R
(n)(d′) + I(d)
)+
E
d0,
where (d′) := (d1, . . . , dn), and
E
d0=
1
2(Hn −
HR
d0)
−1
4≤ I(d) ≤ 0.
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some corollaries (I)
Corollary
With the same notations, we have:
|R(n+1)(d) −
n∏i=1
(n
di + n
)R
(n)(d′)| ≤
1
2log(
d0n
d0 + n) +O(
1
n),
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some Corollaries (II)
The straightforward inductive argument yields
R(n+1)(d) ≤ 1
2
n∑i=0
log
(din
di + n
)+O(1) ≈ 1
2n log(n) +O(1),
and
htu(R(n+1)(d) ) ≤
D(d)
2
(n∑i=0
log
(din
di + n
)+ c
)≈D(d)
2(n log(n) + c).
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some Corollaries (II)
The straightforward inductive argument yields
R(n+1)(d) ≤ 1
2
n∑i=0
log
(din
di + n
)+O(1) ≈ 1
2n log(n) +O(1),
and
htu(R(n+1)(d) ) ≤
D(d)
2
(n∑i=0
log
(din
di + n
)+ c
)≈D(d)
2(n log(n) + c).
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Controlling the growth (in the sense of [Blum-Shub, 86])
Let us consider now the Gaussian distribution γ∆ in H(n+1)(d) induced
by the Bombieri’s norm.
Corollary
With these notations, we have:
Probγ∆
log | Res(n+1)(d) (f0, . . . , fn) |≥ ε−1 +
n∑i=0
∏j 6=i
dj
log ||f ||∆
≤≤D(d)
2(n log(n) + c)ε,
for some constant c > 0.
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some corollaries (III)
* van der Waerden’s U-resultant χU is a classical object inElimination Theory (some times called Chow form, EliminationPolynomial,...).* Upper bounds for the complexity of computing U−resultants wereshown in [Jeronimo-Krick-Sabia-Sombra, 03]
* In [Heintz-Morgenstern, 93]: Computation of the U−resultant isNP–hard.Modestly, we may immediate obtain:
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some corollaries (III)
* van der Waerden’s U-resultant χU is a classical object inElimination Theory (some times called Chow form, EliminationPolynomial,...).* Upper bounds for the complexity of computing U−resultants wereshown in [Jeronimo-Krick-Sabia-Sombra, 03]* In [Heintz-Morgenstern, 93]: Computation of the U−resultant isNP–hard.
Modestly, we may immediate obtain:
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some corollaries (III)
* van der Waerden’s U-resultant χU is a classical object inElimination Theory (some times called Chow form, EliminationPolynomial,...).* Upper bounds for the complexity of computing U−resultants wereshown in [Jeronimo-Krick-Sabia-Sombra, 03]* In [Heintz-Morgenstern, 93]: Computation of the U−resultant isNP–hard.Modestly, we may immediate obtain:
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Some Corollaries (IV)
Let EU be the expected logarithmic Mahler’s measure of theU−resultant with respect to some projective variety determined bythe degree list (d′) := (d1, . . . , dn) :
Corollary
With these notations, we have
ES
(n)
(d′)[m(χU )] ≤
(n∏i=1
di
)(n log(n) + c) ,
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
And something diophantine
Finally, you may use the ideas by Hardy, Mordell, Davenport andothers3 on the equidistribution of polynomial systems with Gaussianrational coefficients of bounded height in the projective space P(Hd)to conclude to conclude that
Corollary
The same bound holds for the expected Mahler’s measure of theU−resultant for random systems with Gaussian rational coefficients ofbounded height and uniform distribution.
3See also [Castro-Montana-P.-San Martin, 02] or [P.-San Martin, 04].Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
And something diophantine
Finally, you may use the ideas by Hardy, Mordell, Davenport andothers3 on the equidistribution of polynomial systems with Gaussianrational coefficients of bounded height in the projective space P(Hd)to conclude to conclude that
Corollary
The same bound holds for the expected Mahler’s measure of theU−resultant for random systems with Gaussian rational coefficients ofbounded height and uniform distribution.
3See also [Castro-Montana-P.-San Martin, 02] or [P.-San Martin, 04].Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Forthcoming Tasks
These bounds are not satisfactory yet.
There must beimprovements and more precise estimates...ongoing research...
* Continue with studies on average properties of MAAG.* Compare unitarily invariant height to other notions of height......* Continue exploration of the over–determined case...the mainproblem.
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Forthcoming Tasks
These bounds are not satisfactory yet. There must beimprovements and more precise estimates...ongoing research...
* Continue with studies on average properties of MAAG.* Compare unitarily invariant height to other notions of height......* Continue exploration of the over–determined case...the mainproblem.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 92: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/92.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Forthcoming Tasks
These bounds are not satisfactory yet. There must beimprovements and more precise estimates...ongoing research...
* Continue with studies on average properties of MAAG.* Compare unitarily invariant height to other notions of height......
* Continue exploration of the over–determined case...the mainproblem.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 93: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/93.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Forthcoming Tasks
These bounds are not satisfactory yet. There must beimprovements and more precise estimates...ongoing research...
* Continue with studies on average properties of MAAG.* Compare unitarily invariant height to other notions of height......* Continue exploration of the over–determined case...the mainproblem.
Luis M. Pardo MAAGA, Mike’s May 68
![Page 94: Metric Aspects in Algebraic Geometry, on the Average ... · (Celebrating the work of Mike Shub) Luis M. Pardo1 May, 2012 1Univ. de Cantabria. Luis M. Pardo MAAGA, Mike’s May 68](https://reader036.vdocuments.site/reader036/viewer/2022081406/5f0d29bd7e708231d438fbf8/html5/thumbnails/94.jpg)
Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Forthcoming Tasks
Happy May’68, Mike!
and thanks to all of you for your patience!
Luis M. Pardo MAAGA, Mike’s May 68
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Plan of the TalkExpected Growth of Polynomials
Expected Minimum SeparationExpected Distance between two Complete Intersections
On the Height of the Multi-variate Resultant Variety
Forthcoming Tasks
Happy May’68, Mike!
and thanks to all of you for your patience!
Luis M. Pardo MAAGA, Mike’s May 68