elimination theory in the 21st centuryfaculty.tcu.edu/gfriedman/cbms2018/andrea.pdfelimination...

Elimination Theory in the 21st century

Carlos D’Andrea

NSF-CBMS Conference on Applications of Polynomial Systems

Carlos D’Andrea

Before we start...

Session on Open ProblemsFriday 3.30 pm

5 minutes per presentation

to book a time slot, contact Hal or me

Carlos D’Andrea

Before we start...

Carlos D’Andrea

Before we start...

Carlos D’Andrea

Before we start...

Carlos D’Andrea

Before we start...

Carlos D’Andrea

The Paradigm of the 21st Century

Apart from being crazy aboutapplications and implementations...

Carlos D’Andrea

Apart from being crazy aboutapplications

and implementations...

Carlos D’Andrea

“Cost” of Elimination

The number and “size” of the solutions ofF1(X1, . . . ,Xn) = 0F2(X1, . . . ,Xn) = 0

......

...Fn(X1, . . . ,Xn) = 0

where “size” of Fi = (d , L)is bounded by and generically equal to(

dn, ndn−1L)

Carlos D’Andrea

......

...Fn(X1, . . . ,Xn) = 0

dn, ndn−1L)

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......

...Fn(X1, . . . ,Xn) = 0

where “size” of Fi = (d , L)

is bounded by and generically equal to(dn, ndn−1L

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......

...Fn(X1, . . . ,Xn) = 0

dn, ndn−1L)

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The output is already exponential!!!

Moreover: the complexity ofcomputing Grobner bases is double

exponential in (d , L)

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Moreover:

the complexity ofcomputing Grobner bases is double

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Moreover: the complexity ofcomputing Grobner bases is double

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What can we do then???

“Relax” the input (and expect theoutput to be “relaxed” too!)

Change the Computational Model

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“Relax” the input

(and expect theoutput to be “relaxed” too!)

Carlos D’Andrea

Change the Computational ModelCarlos D’Andrea

Relaxing the input

Tropical Geometry“Relaxed” resultantsImplicitization matrices(Wednesday)Syzygies and Rees Algebras(Wednesday) . . .

Carlos D’Andrea

Relaxing the input

Tropical Geometry

“Relaxed” resultantsImplicitization matrices(Wednesday)Syzygies and Rees Algebras(Wednesday) . . .

Carlos D’Andrea

Relaxing the input

Tropical Geometry“Relaxed” resultants

Implicitization matrices(Wednesday)Syzygies and Rees Algebras(Wednesday) . . .

Carlos D’Andrea

Relaxing the input

Tropical Geometry“Relaxed” resultantsImplicitization matrices

(Wednesday)Syzygies and Rees Algebras(Wednesday) . . .

Carlos D’Andrea

Relaxing the input

Tropical Geometry“Relaxed” resultantsImplicitization matrices(Wednesday)

Syzygies and Rees Algebras(Wednesday) . . .

Carlos D’Andrea

Relaxing the input

Tropical Geometry“Relaxed” resultantsImplicitization matrices(Wednesday)Syzygies and Rees Algebras

(Wednesday) . . .

Carlos D’Andrea

Relaxing the input

Tropical Geometry“Relaxed” resultantsImplicitization matrices(Wednesday)Syzygies and Rees Algebras(Wednesday)

Carlos D’Andrea

Relaxing the input

Tropical Geometry“Relaxed” resultantsImplicitization matrices(Wednesday)Syzygies and Rees Algebras(Wednesday) . . .

Carlos D’Andrea

Changing the model

Probabilistic algorithmsComputations “over the Reals”“Parallelization”Homotopy methods (Tomorrow). . .

Carlos D’Andrea

Changing the model

Probabilistic algorithms

Computations “over the Reals”“Parallelization”Homotopy methods (Tomorrow). . .

Carlos D’Andrea

Changing the model

Probabilistic algorithmsComputations “over the Reals”

“Parallelization”Homotopy methods (Tomorrow). . .

Carlos D’Andrea

Changing the model

Probabilistic algorithmsComputations “over the Reals”“Parallelization”

Homotopy methods (Tomorrow). . .

Carlos D’Andrea

Changing the model

Probabilistic algorithmsComputations “over the Reals”“Parallelization”Homotopy methods

(Tomorrow). . .

Carlos D’Andrea

Changing the model

Probabilistic algorithmsComputations “over the Reals”“Parallelization”Homotopy methods (Tomorrow)

Carlos D’Andrea

Changing the model

Probabilistic algorithmsComputations “over the Reals”“Parallelization”Homotopy methods (Tomorrow). . .

Carlos D’Andrea

Some “Tapas” into the 21st century

Disclaimer

These are not the highlights of the 21stcentury (so far), just a sampling of resultsAll these areas are very active and promisingright now, more will appear in the future...A lot of important names and interesting areaswill be omitted...sorry!

Carlos D’Andrea

Disclaimer

Carlos D’Andrea

Disclaimer

Carlos D’Andrea

Disclaimer

These are not the highlights of the 21stcentury (so far), just a sampling of results

All these areas are very active and promisingright now, more will appear in the future...A lot of important names and interesting areaswill be omitted...sorry!

Carlos D’Andrea

Disclaimer

These are not the highlights of the 21stcentury (so far), just a sampling of resultsAll these areas are very active and promisingright now

, more will appear in the future...A lot of important names and interesting areaswill be omitted...sorry!

Carlos D’Andrea

Disclaimer

These are not the highlights of the 21stcentury (so far), just a sampling of resultsAll these areas are very active and promisingright now, more will appear in the future...

A lot of important names and interesting areaswill be omitted...sorry!

Carlos D’Andrea

Disclaimer

These are not the highlights of the 21stcentury (so far), just a sampling of resultsAll these areas are very active and promisingright now, more will appear in the future...A lot of important names and interesting areaswill be omitted...

sorry!

Carlos D’Andrea

Disclaimer

Carlos D’Andrea

Tropical: from Exponential to Polynomial

The logarithmic limit set of avariety V ⊂ Cn is a fan in Rn, the

Tropical Variety of V(MSC 2010 14Txx)

Carlos D’Andrea

The logarithmic limit set of avariety V ⊂ Cn is a fan in Rn

, theTropical Variety of V

(MSC 2010 14Txx)

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Tropical Variety of V

(MSC 2010 14Txx)

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Tropical Variety of V(MSC 2010 14Txx)

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Computing Tropical Varieties

is “fast”

encodes a lot of information aboutthe original variety (dimension,degree, singularities...)

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is “fast”

Carlos D’Andrea

is “fast”

encodes a lot of information aboutthe original variety

(dimension,degree, singularities...)

Carlos D’Andrea

is “fast”

Carlos D’Andrea

First tapa: Tropical Discriminants

Alicia Dickenstein, Eva MariaFeitchner, Bernd Sturmfels“Tropical Discriminants”

( JAMS 2007)

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First tapa: Tropical Discriminants

Alicia Dickenstein, Eva MariaFeitchner, Bernd Sturmfels“Tropical Discriminants”

( JAMS 2007)

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Main Result

Theorem 1.1For any d × n matrix A, the tropicalA-discriminant τ (XA∗) equals the

Minkowski sum of theco-Bergman fan B∗(A) and the

row space of A.

Carlos D’Andrea

Main Result

Theorem 1.1

For any d × n matrix A, the tropicalA-discriminant τ (XA∗) equals the

row space of A.

Carlos D’Andrea

Main Result

Theorem 1.1For any d × n matrix A, the tropicalA-discriminant τ (XA∗) equals the

row space of A.

Carlos D’Andrea

What is the co-Bergman fan?

“is” the tropicalization of the kernelof A

Carlos D’Andrea

What is the co-Bergman fan?

“is” the tropicalization of the kernelof A

Carlos D’Andrea

Tapa # 2:Tropical Elimination

Bernd Sturmfels, Jenia Tevelev“Elimination Theory for Tropical

Varieties”( MRL 2008)

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Tapa # 2:Tropical Elimination

Bernd Sturmfels, Jenia Tevelev“Elimination Theory for Tropical

Varieties”( MRL 2008)

Carlos D’Andrea

Theorem1.1

Let α : Tn → Td be a homomorphism of tori and letA : Zn → Zd be the corresponding linear map of

lattices of one-parameter subgroups. Suppose thatα induces a generically finite morphism from X ontoα(X ). Then A induces a generically finite map oftropical varieties from T (X ) onto T (α(X )).

Carlos D’Andrea

Theorem1.1

Let α : Tn → Td be a homomorphism of tori

and letA : Zn → Zd be the corresponding linear map of

Carlos D’Andrea

Theorem1.1

lattices of one-parameter subgroups.

Suppose thatα induces a generically finite morphism from X ontoα(X ). Then A induces a generically finite map oftropical varieties from T (X ) onto T (α(X )).

Carlos D’Andrea

Theorem1.1

lattices of one-parameter subgroups. Suppose thatα induces a generically finite morphism from X ontoα(X ).

Then A induces a generically finite map oftropical varieties from T (X ) onto T (α(X )).

Carlos D’Andrea

Theorem1.1

Carlos D’Andrea

What about Tropical Resultants?

Anders Jensen, Josephine Yu“Computing Tropical Resultants”

(JA 2013)

Carlos D’Andrea

What about Tropical Resultants?

Anders Jensen, Josephine Yu“Computing Tropical Resultants”

(JA 2013)

Carlos D’Andrea

From the abstract:

We fix the supports A = (A1, . . . ,Ak)of a list of tropical polynomials

and define the tropical resultantTR(A) to be the set of choices ofcoefficients such that the tropical

polynomials have a common solution.

Carlos D’Andrea

From the abstract:

We fix the supports A = (A1, . . . ,Ak)of a list of tropical polynomials

and define the tropical resultantTR(A) to be the set of choices ofcoefficients such that the tropical

polynomials have a common solution.

Carlos D’Andrea

From the abstract II

We prove that TR(A) is thetropicalization of the algebraic variety

of solvable systems and that itsdimension can be computed in

polynomial time...

Carlos D’Andrea

From the abstract II

We prove that TR(A) is thetropicalization of the algebraic variety

of solvable systems and that itsdimension can be computed in

polynomial time...

Carlos D’Andrea

From the abstract III

We also present a new algorithm forrecovering a Newton polytopefrom the support of its tropical

hypersurface. We use this to computethe Newton polytope of the sparse

resultant polynomial in the case whenTR(A) is of codimension 1...

Carlos D’Andrea

“Relaxed” resultants

In the “real world” systems ofequations are neither homogeneousnor all the monomials appear in the

expansion F0 = a01 + a02X12 X2

2 + a03X1 X23

F1 = a10 + a11X12 + a12X1 X2

F2 = a20X13 + a21X1 X2

Carlos D’Andrea

“Relaxed” resultants

In the “real world” systems ofequations are neither homogeneousnor all the monomials appear in the

expansion F0 = a01 + a02X12 X2

2 + a03X1 X23

F1 = a10 + a11X12 + a12X1 X2

F2 = a20X13 + a21X1 X2

Carlos D’Andrea

Sparse Resultants (after GKZ)

D, Martın Sombra“A Poisson formula for the sparse resultant”

(PLMS 2015)

W = {(ci ,a, ξ) : Fi(ξ) = 0∀i} ⊂ PA0 × . . .× PAn × (C×)n

↓ ↓ ππ(W ) ⊂ PA0 × . . .× PAn

The sparse resultant of F0, . . . ,Fn is the definingequation of the direct image π∗Wdn 7→ MV (N(A1), . . . ,N(An))

ndn−1L 7→ nMV (N(A1), . . . ,N(An))L

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(PLMS 2015)

↓ ↓ ππ(W ) ⊂ PA0 × . . .× PAn

ndn−1L 7→ nMV (N(A1), . . . ,N(An))L

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(PLMS 2015)

↓ ↓ ππ(W ) ⊂ PA0 × . . .× PAn

The sparse resultant of F0, . . . ,Fn is the definingequation of the direct image π∗W

dn 7→ MV (N(A1), . . . ,N(An))ndn−1L 7→ nMV (N(A1), . . . ,N(An))L

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(PLMS 2015)

↓ ↓ ππ(W ) ⊂ PA0 × . . .× PAn

ndn−1L 7→ nMV (N(A1), . . . ,N(An))L

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(PLMS 2015)

↓ ↓ ππ(W ) ⊂ PA0 × . . .× PAn

ndn−1L 7→ nMV (N(A1), . . . ,N(An))L

Carlos D’Andrea

Other “relaxed” resultants

By replacing (C×)n with other “efficient” varietiesX , we get “efficient” resultants:

W = {(ci ,a, ξ) : Fi(ξ) = 0 ∀i} ⊂ PN0 × . . .× PNn × X↓ ↓ π

π(W ) ⊂ PN0 × . . .× PNn

Carlos D’Andrea

Other “relaxed” resultants

By replacing (C×)n with other “efficient” varietiesX , we get “efficient” resultants:

W = {(ci ,a, ξ) : Fi(ξ) = 0 ∀i} ⊂ PN0 × . . .× PNn × X↓ ↓ π

π(W ) ⊂ PN0 × . . .× PNn

Carlos D’Andrea

Examples of “relaxed” resultants

Reduced Resultants (Zariski,Jouanolou,...)

Weighted Resultants (Jouanolou)

Parametric Resultants(Buse-Elkadi-Mourrain)

Residual Resultants (Buse)

Carlos D’Andrea

. . .Carlos D’Andrea

Implicitization matrices

Recall van der Waerden’s matrix introduced thismorning:

Ms(λ)

xβ...

xαFi...

For s � 0, the rank of Ms(λ) drops iff there is a

common solution

Carlos D’Andrea

Ms(λ)

xβ...

xαFi...

For s � 0, the rank of Ms(λ) drops iff there is acommon solution

Carlos D’Andrea

Ms(λ)

xβ...

xαFi...

For s � 0, the rank of Ms(λ) drops iff there is a

common solutionCarlos D’Andrea

“Rank” vs “Determinant”

Laurent Buse“Implicit matrix representations of

rational Bezier curves and surfaces”(CAGD 2014)

Carlos D’Andrea

“Rank” vs “Determinant”

Laurent Buse“Implicit matrix representations of

rational Bezier curves and surfaces”(CAGD 2014)

Carlos D’Andrea

Given a rational parametrization of aspatial curve (φ1(t), φ2(t), φ3(t))

Classic implicitization: Compute theequations Fi(X ,Y ,Z ) of the image

21st century implicitization: Computea matrix M(X ,Y ,Z ) such that its

rank drops on the points of the curve

Carlos D’Andrea

Classic implicitization:

Compute theequations Fi(X ,Y ,Z ) of the image

Carlos D’Andrea

21st century implicitization:

Computea matrix M(X ,Y ,Z ) such that its

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rank drops on the points of the curveCarlos D’Andrea

Properties

Entries are linear in X ,Y ,ZTesting rank is fast by usingnumerical methodsOver the reals one can use SVDand other numerical methodsWell suited for a lot of problems inCAGD (properness, inversion,...)

Carlos D’Andrea

Properties

Entries are linear in X ,Y ,Z

Testing rank is fast by usingnumerical methodsOver the reals one can use SVDand other numerical methodsWell suited for a lot of problems inCAGD (properness, inversion,...)

Carlos D’Andrea

Properties

Entries are linear in X ,Y ,ZTesting rank is fast by usingnumerical methods

Over the reals one can use SVDand other numerical methodsWell suited for a lot of problems inCAGD (properness, inversion,...)

Carlos D’Andrea

Properties

Entries are linear in X ,Y ,ZTesting rank is fast by usingnumerical methodsOver the reals one can use SVDand other numerical methods

Well suited for a lot of problems inCAGD (properness, inversion,...)

Carlos D’Andrea

Properties

Entries are linear in X ,Y ,ZTesting rank is fast by usingnumerical methodsOver the reals one can use SVDand other numerical methodsWell suited for a lot of problems inCAGD (properness, inversion,...)

Carlos D’Andrea

More about this on Wednesday!

Carlos D’Andrea

Syzygies and Rees Algebras

Warming up tapa

The rest on Wednesday :-)

Carlos D’Andrea

Warming up tapa

Carlos D’Andrea

Warming up tapa

Carlos D’Andrea

Example (Sederberg & Chen 1995)

The implicit equation of a rational quartic can becomputed as a 2× 2 determinant.

If the curve has a triple point, then one row islinear and the other is cubic.

Otherwise, both rows are quadratic.

Carlos D’Andrea

A quartic with a triple point

φ(t0, t1) = (t40 − t41 : −t20 t21 : t0t31 )

F (X0,X1,X2) = X 42 − X 4

1 − X0X1X22

L1,1(T ,X ) = T0X2 + T1X1

L1,3(T ,X ) = T0(X 31 + X0X

22 ) + T1 X

X 31 + X0X

22 X 3

Carlos D’Andrea

φ(t0, t1) = (t40 − t41 : −t20 t21 : t0t31 )

F (X0,X1,X2) = X 42 − X 4

1 − X0X1X22

L1,1(T ,X ) = T0X2 + T1X1

L1,3(T ,X ) = T0(X 31 + X0X

22 ) + T1 X

X 31 + X0X

22 X 3

Carlos D’Andrea

φ(t0, t1) = (t40 − t41 : −t20 t21 : t0t31 )

F (X0,X1,X2) = X 42 − X 4

1 − X0X1X22

L1,1(T ,X ) = T0X2 + T1X1

L1,3(T ,X ) = T0(X 31 + X0X

22 ) + T1 X

X 31 + X0X

22 X 3

Carlos D’Andrea

φ(t0, t1) = (t40 − t41 : −t20 t21 : t0t31 )

F (X0,X1,X2) = X 42 − X 4

1 − X0X1X22

L1,1(T ,X ) = T0X2 + T1X1

L1,3(T ,X ) = T0(X 31 + X0X

22 ) + T1 X

X 31 + X0X

22 X 3

)Carlos D’Andrea

A quartic without triple points

φ(t0 : t1) = (t40 : 6t20 t21 − 4t41 : 4t30 t1 − 4t0t

F (X ) = X 42 + 4X0X

31 + 2X0X1X

22 −16X 2

0X21 −6X 2

0X22 + 16X 3

L1,2(T ,X ) = T0(X1X2 − X0X2) + T1(−X 22 − 2X0X1 + 4X 2

L1,2(T ,X ) = T0(X 21 + 1

2X 22 − 2X0X1) + T1(X0X2 − X1X2)

Carlos D’Andrea

φ(t0 : t1) = (t40 : 6t20 t21 − 4t41 : 4t30 t1 − 4t0t

F (X ) = X 42 + 4X0X

31 + 2X0X1X

22 −16X 2

0X21 −6X 2

0X22 + 16X 3

L1,2(T ,X ) = T0(X1X2 − X0X2) + T1(−X 22 − 2X0X1 + 4X 2

L1,2(T ,X ) = T0(X 21 + 1

2X 22 − 2X0X1) + T1(X0X2 − X1X2)

Carlos D’Andrea

φ(t0 : t1) = (t40 : 6t20 t21 − 4t41 : 4t30 t1 − 4t0t

F (X ) = X 42 + 4X0X

31 + 2X0X1X

22 −16X 2

0X21 −6X 2

0X22 + 16X 3

L1,2(T ,X ) = T0(X1X2 − X0X2) + T1(−X 22 − 2X0X1 + 4X 2

L1,2(T ,X ) = T0(X 21 + 1

2X 22 − 2X0X1) + T1(X0X2 − X1X2)

Carlos D’Andrea

A very compact formula

If the curve has a point of multiplicity d − 1the implicit equation is always a 2× 2 determinantof a moving line and a moving curve of degree

d − 1∣∣∣∣ L1,1(X ) L′1,1(X )

L1,d−1(X ) L′1,d−1(X )

∣∣∣∣

Carlos D’Andrea

If the curve has a point of multiplicity d − 1

the implicit equation is always a 2× 2 determinantof a moving line and a moving curve of degree

d − 1∣∣∣∣ L1,1(X ) L′1,1(X )

L1,d−1(X ) L′1,d−1(X )

∣∣∣∣

Carlos D’Andrea

If the curve has a point of multiplicity d − 1the implicit equation is always a 2× 2 determinantof a moving line and a moving curve of degree

d − 1∣∣∣∣ L1,1(X ) L′1,1(X )

L1,d−1(X ) L′1,d−1(X )

∣∣∣∣Carlos D’Andrea

The “method” of moving curves

The implicit equation of a rational curve should becomputed as the determinant of a small matrix

whose entries are∣∣∣∣∣∣∣∣some moving lines

some moving conicssome moving cubics

· · ·

∣∣∣∣∣∣∣∣the more singular the curve, the simpler the

description of the determinant

Carlos D’Andrea

whose entries are

∣∣∣∣∣∣∣∣some moving lines

· · ·

description of the determinant

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· · ·

∣∣∣∣∣∣∣∣

the more singular the curve, the simpler thedescription of the determinant

Carlos D’Andrea

· · ·

description of the determinantCarlos D’Andrea

In general, we do not know..

∣∣∣∣∣∣∣∣which moving lines?

which moving conics?which moving cubics?

· · ·

∣∣∣∣∣∣∣∣

Carlos D’Andrea

In general, we do not know..

∣∣∣∣∣∣∣∣which moving lines?

which moving conics?which moving cubics?

· · ·

∣∣∣∣∣∣∣∣

Carlos D’Andrea

Some order in the 21st century

David Cox “The moving curve ideal and the Reesalgebra” (TCS 2008)

Kφ := {Moving curves following φ} =homogeneous elements in the kernel of

K[T0,T1,X0,X1,X2] → K[T0,T1, s]Ti 7→ Ti

X0 7→ φ0(T )sX1 7→ φ1(T )sX2 7→ φ2(T )s

The image of this map is the Rees Algebra of φ

Carlos D’Andrea

K[T0,T1,X0,X1,X2] → K[T0,T1, s]Ti 7→ Ti

X0 7→ φ0(T )sX1 7→ φ1(T )sX2 7→ φ2(T )s

Carlos D’Andrea

K[T0,T1,X0,X1,X2] → K[T0,T1, s]Ti 7→ Ti

X0 7→ φ0(T )sX1 7→ φ1(T )sX2 7→ φ2(T )s

Carlos D’Andrea

K[T0,T1,X0,X1,X2] → K[T0,T1, s]Ti 7→ Ti

X0 7→ φ0(T )sX1 7→ φ1(T )sX2 7→ φ2(T )s

The image of this map is the Rees Algebra of φCarlos D’Andrea

Method of moving curves revisited

The implicit equation should be obtained as thedeterminant of a matrix with∣∣∣∣∣∣∣∣

· · ·some minimal generators of Kφ

and relations among them· · ·

∣∣∣∣∣∣∣∣The more singular the curve, the “simpler” the

description of Kφ

Carlos D’Andrea

The implicit equation should be obtained as thedeterminant of a matrix with

∣∣∣∣∣∣∣∣· · ·

some minimal generators of Kφand relations among them

· · ·

description of Kφ

Carlos D’Andrea

∣∣∣∣∣∣∣∣

The more singular the curve, the “simpler” thedescription of Kφ

Carlos D’Andrea

description of KφCarlos D’Andrea

21st Century Problem

Compute a minimal system ofgenerators of Kφ for any φ

We are working on that!(More on Wednesday)

Carlos D’Andrea

Compute a minimal system ofgenerators of Kφ

for any φ

Carlos D’Andrea

We are working on that!

(More on Wednesday)

Carlos D’Andrea

Changing the model of computation

Carlos D’Andrea

Probabilistic Algorithms: faster they are

Gabriela Jeronimo, Teresa Krick,Juan Sabia, Martın Sombra

“The Computational Complexity ofthe Chow Form”( JFoCM 2004)

Carlos D’Andrea

Probabilistic Algorithms: faster they are

Gabriela Jeronimo, Teresa Krick,Juan Sabia, Martın Sombra

“The Computational Complexity ofthe Chow Form”( JFoCM 2004)

Carlos D’Andrea

From the abstract:

We present a bounded probability algorithm forthe computation of the Chow forms (resultants) of

the equidimensional components of a variety...

The expected complexity of the algorithm ispolynomial in the size and the geometric degree of

the input equation system defining the variety...

As an application, we obtain an algorithm tocompute a subclass of sparse resultants, whose

complexity is polynomial in the dimension and thevolume of the input set of exponents...

Carlos D’Andrea

From the abstract:

Carlos D’Andrea

From the abstract:

Carlos D’Andrea

Main idea

Given (equations) of a d-dimensional varietyV ⊂ Pn

Cut V with “general” (probabilistically) dhyperplanes to obtain finite points:V ∩ {x1 = . . . = xd = 0} = {ξ1, . . . , ξD}From (a geometric resolution of) {ξ1, . . . , ξD},lift (Newton’s Method) to the Chow Form ofV ∩ {x1 = . . . = xd−1 = 0}Do it (d − 1) times more...

Carlos D’Andrea

Main idea

Carlos D’Andrea

Main idea

Cut V with “general” (probabilistically) dhyperplanes to obtain finite points:V ∩ {x1 = . . . = xd = 0} = {ξ1, . . . , ξD}From (a geometric resolution of) {ξ1, . . . , ξD},lift (Newton’s Method) to the Chow Form ofV ∩ {x1 = . . . = xd−1 = 0}

Do it (d − 1) times more...

Carlos D’Andrea

Main idea

Carlos D’Andrea

Star Model: Straight-Line Programs

Gabriela Jeronimo, Juan Sabia“Sparse resultants and straight-line

programs”(JSC 2018)

Carlos D’Andrea

Star Model: Straight-Line Programs

Gabriela Jeronimo, Juan Sabia“Sparse resultants and straight-line

programs”(JSC 2018)

Carlos D’Andrea

What is a straight-line program?

It is a program with no branches,no loops, no conditional

statements, nocomparisons...just a sequence of

basic operations

Carlos D’Andrea

It is a program with no branches

,no loops, no conditional

basic operations

Carlos D’Andrea

It is a program with no branches,no loops

, no conditionalstatements, no

comparisons...just a sequence ofbasic operations

Carlos D’Andrea

statements

, nocomparisons...just a sequence of

basic operations

Carlos D’Andrea

statements, nocomparisons

...just a sequence ofbasic operations

Carlos D’Andrea

basic operationsCarlos D’Andrea

From the abstract:

We prove that the sparse resultant, redefined byD’Andrea and Sombra and by Esterov as a power ofthe classical sparse resultant, can be evaluated in anumber of steps which is polynomial in its degree,

its number of variables and the size of theexponents ...

Moreover, we design a probabilistic algorithm ofthis order of complexity to compute a straight-lineprogram that evaluates it within this number of

steps.

Carlos D’Andrea

From the abstract:

steps.

Carlos D’Andrea

From the abstract:

steps.

Carlos D’Andrea

Main ideas

SLP’s

Geometric Resolution

Newton Hensel Lifting

Pade Approximations...

Carlos D’Andrea

Main ideas

SLP’s

Carlos D’Andrea

Main ideas

SLP’s

Carlos D’Andrea

Main ideas

SLP’s

Carlos D’Andrea

Main ideas

SLP’s

Carlos D’Andrea

Anoter “real” model: BSS-machine

is a Random Access Machinewith registers that can store

arbitrary real numbers and thatcan compute rational functions over

the reals at unit cost

Carlos D’Andrea

is a Random Access Machine

with registers that can storearbitrary real numbers and thatcan compute rational functions over

Carlos D’Andrea

arbitrary real numbers

and thatcan compute rational functions over

Carlos D’Andrea

arbitrary real numbers and thatcan compute rational functions over

the reals at unit costCarlos D’Andrea

Parallelize: find just one solution

Given f (x) ∈ Z[x ], compute a smallapproximate root, fast

Carlos D’Andrea

Given f (x) ∈ Z[x ], compute a small

approximate root, fast

Carlos D’Andrea

Given f (x) ∈ Z[x ], compute a smallapproximate root

, fast

Carlos D’Andrea

Given f (x) ∈ Z[x ], compute a smallapproximate root, fast

Carlos D’Andrea

What is an approximate root?

Carlos D’Andrea

Steve Smale’s 17th’s problem (1998)

Is there an algorithm which computesan approximate solution of asystem of polynomials in time

polynomial on the average, inthe size of the input ?

Carlos D’Andrea

Steve Smale’s 17th’s problem (1998)

Is there an algorithm which computesan approximate solution of asystem of polynomials in time

polynomial on the average, inthe size of the input ?

Carlos D’Andrea

Solving Smale’s 17th Problem

Carlos Beltran, Luis Miguel Pardo. “On Smale’s17th Problem: A Probabilistic Positive answer”( JFoCM 2008)

Carlos Beltran, Luis Miguel Pardo. “Smale’s17th Problem: Average Polynomial Time tocompute affine and projective solutions” (JAMS2009)

Felipe Cucker, Peter Burgisser. “On a problemposed by Steve Smale” (Ann.Math 2011)

Carlos D’Andrea

Pierre Lairez (2017)

“A deterministic algorithm tocompute approximate roots of

polynomial systems in polynomialaverage time”JFoCM (2017)

Carlos D’Andrea

Homotopies (In depth tomorrow!)

Start with an “easy” system“Chase” the roots with anhomotopy + Newton’s method

Carlos D’Andrea

Start with an “easy” system

“Chase” the roots with anhomotopy + Newton’s method

Carlos D’Andrea

Start with an “easy” system“Chase” the roots with anhomotopy + Newton’s method

Carlos D’Andrea

Numerical Algebraic Geometry

By using homotopies, you cancompute very fast

Components

Multiplicities

Decomposition

Dimension

Carlos D’Andrea

Components

Multiplicities

Decomposition

Dimension

Carlos D’Andrea

Components

Multiplicities

Decomposition

Dimension

Carlos D’Andrea

Components

Multiplicities

Decomposition

Dimension

Carlos D’Andrea

Components

Multiplicities

Decomposition

Dimension

Carlos D’Andrea

Components

Multiplicities

Decomposition

Dimension

Carlos D’Andrea

Components

Multiplicities

Decomposition

Dimension

. . .Carlos D’Andrea

Not in the menu today...

Differential/Difference EliminationReal EliminationApplicationsImplementations. . .

Carlos D’Andrea

Differential/Difference Elimination

Real EliminationApplicationsImplementations. . .

Carlos D’Andrea

Differential/Difference EliminationReal Elimination

ApplicationsImplementations. . .

Carlos D’Andrea

Differential/Difference EliminationReal EliminationApplications

Implementations. . .

Carlos D’Andrea

Differential/Difference EliminationReal EliminationApplicationsImplementations

Carlos D’Andrea

Differential/Difference EliminationReal EliminationApplicationsImplementations. . .

Carlos D’Andrea

Conclusion

The 21st century is/will be driven byApplicationsFast & efficient computations

Interesting challenges and problemsneed your help here!

Carlos D’Andrea

Conclusion

The 21st century is/will be driven byApplications

Fast & efficient computationsInteresting challenges and problems

need your help here!

Carlos D’Andrea

Conclusion

Carlos D’Andrea

Conclusion

Carlos D’Andrea

Thanks!

http://www.ub.edu/arcades/cdandrea.html

Carlos D’Andrea

elimination theory in the 21st centuryfaculty.tcu.edu/gfriedman/cbms2018/andrea.pdfelimination...

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