Numerical Algebraic Geometryvia Macaulay’s Perspective

Barry H. Dayton

Northeastern Illinois UniversityChicago, Illinois

bhdayton@neiu.edu

Numerical Methods for Polynomial SystemsSIAM AN12, July 2012

Minneapolis

Motivation

[A solution] might be regarded as in some measure complete if itwere admitted that [the] problem is finished with when its solutionhas been reduced to a finite number of feasible operations. Ifhowever the operations are too numerous or too involved to becarried out in practice the solution is only a theoretical one; and itsimportance then lies not in itself, but in the theorems with which itis associated and to which it leads. Such a theoretical solutionmust be regarded as preliminary and not the final stage ofthe consideration of the problem.

F.S. Macaulay, The Algebraic Theory of Modular Systems, 1916

We will be using techniques developed or mentioned in Macaulay’s1916 book such as H-bases, Macaulay (dialytic) arrays Sylvester(resultant) arrays and duals (inverse functions and arrays).

H-bases

An H-basis was defined in 1916 by Macaulay as follows:

The distinctive property of an H-basis (F1,F2, . . . ,Fk)of M is than any member F of M can be put in the formA1F1 + A2F2 + · · ·+ AkFk where AiFi (i = 1, 2, . . . , k) isnot of greater degree than F . Every module [ideal] hasan H-basis, which may necessarily consist of moremembers than would suffice for a basis in general

Note that a homogeneous basis for a homogeneous ideal is anH-basis, the original motivation for the name H-basis

We will see below that H-bases are associated with maximal rankSylvester Matrices.

H-bases appear to be an appropriate alternative to Grobner basesin the numerical case.

Local Dual Functionals

A Local dual functional is a C-linear map

C[[x]]/C[[x]]I∣∣∣x−→ C

Here C[[x]]/C[[x]]I∣∣∣x

denotes the local ring at point x.

A typical such functional is given by a finite sum

∑|j|<n

βk∂xj [x], where ∂xj [x] ≡ 1

j1! · · · js !

∂j1+···+js

∂x j11 · · · ∂x js

s

∣∣∣∣∣x

These have been studied by Max Noether, Lasker, Macaulay,Grobner, Mora, Moller, Mourrain, Stetter, L.Zhi et. al., and [?]among others.

Global dual functionals

A Global dual functional, in my terminology, is a C-linear map

C[x]/I −→ C

Let G(A) denote the C-vector space of global duals of A.

Example: Let A = C[x] = C[x1, . . . , xs ]. Then G(A) can beidentified with the C-vector space C[[X1, · · · ,Xs ]] of formal powerseries where, for Xk = X k1

1 · · ·X kss and xj = x j1

1 . . . xjss ,

Xk(xj)

=

{1 if j = k,

0 if j 6= k.

More generally if A = C[x]/I then G(A) is the subspace ofG(C[x]) given by G(A) = {d | d(f ) = 0 for all f ∈ I}

Difference between local and global duals

I Global duals are based at the origin regardless of whether theorigin is in V (I). Global duals are series and will berepresented by Macaulay matrices. Rings of analytic functionsdo not have global duals.

I Local duals are based at some point of V (I), each point hasa separate space of local duals. Local duals are finite sumsand represented by Sylvester matrices. Analytic function ringsdo have local duals, see [?].

Note: If A is an affine ring with homogeneous basis then theMacaulay and Sylvester arrays are row equivalent. In this specialcase the local dual at the origin agrees with the global dual. Onecan use this fact to find the global dual of a homogeneous variety.

The Functorial Property of global duals

Suppose algebraic sets X ⊆ As , I = I (X ) and Y ⊆ Ar , J = I (Y)and

φ : As −→ Ar

is an affine map φ = [f1, . . . , fr ] where fi : C[x1, . . . , xs ]→ C arepolynomial functions such that φ(X ) ⊆ Y. This gives a ring map

φ∗ : C[y1, . . . , yr ]/J −→ C[x1, . . . , xs ]/I

given by φ∗(g) = g(f1, . . . , fr ). Then we get a linear map

φ∗ : G(C[x]/I) −→ G(C[y]/J )

defined by φ∗(d) = d ◦ φ∗.

It is seen that we have a covariant functor from algebraic sets Xand affine maps to C vector spaces and linear maps.

Key Algorithms

1. Basis + point → Local Dual There are many versions ofthis standard algorithm, Macaulay’s and our Macaulay matrixversion [?] as well as closedness methods by Mourrain, Zeng,Zhi et. al., Hao-Sommese-Zeng etc.

2. Local duals → Global Dual aka local-global method.Calculates global dual using local dual structure at severalpoints on a component. [?, ?]

3. H-basis → Global dual

4. Global dual of X → Global dual of φ(X ) (FunctorialProperty)

5. Global Dual → Minimal H-basis See [?] etc.

Some more details on my versions later in this talk.

Applications: Extrinsic vs. Intrinsic representationExtrinsic Intrinsic method

Implicit description ofcomponent of algebraicset

Witness points definingcomponent

local-global ⇐

Implicit representationof curve or surface

parametric representa-tion

functorality ⇔

Implict representationof curve, possibly notcomplete intersection

plot of curve functorality ⇒

Simple Example of Parameterizing Curve

Consider the Bow curve f = x4 − x2y + y 3.

Using Mathematica

fh = x4 − x2yz + y 3zfh = x4 − x2yz + y 3zfh = x4 − x2yz + y 3z

A = {{1, 0, 0}, {0, 0, 1}, {0, 1, 0}};A = {{1, 0, 0}, {0, 0, 1}, {0, 1, 0}};A = {{1, 0, 0}, {0, 0, 1}, {0, 1, 0}};(* put singular point at (0, 1, 0) in P2 *)

G = HB2GD[{fh}, 6, {x , y , z}];G = HB2GD[{fh}, 6, {x , y , z}];G = HB2GD[{fh}, 6, {x , y , z}];H = GMap[G,A.{x , y , z}, 6, {x , y , z}, {x , y , z}];H = GMap[G,A.{x , y , z}, 6, {x , y , z}, {x , y , z}];H = GMap[G,A.{x , y , z}, 6, {x , y , z}, {x , y , z}];B = MBasis[H, 4, {x , y , z}, 1.*∧-12];B = MBasis[H, 4, {x , y , z}, 1.*∧-12];B = MBasis[H, 4, {x , y , z}, 1.*∧-12];

g = B[[1]]/.{z → 1, x → t}g = B[[1]]/.{z → 1, x → t}g = B[[1]]/.{z → 1, x → t}t4 + y − t2y (* so y = t4

−1+t2*)

gf = (y/.Solve[g == 0, {y}])[[1]];gf = (y/.Solve[g == 0, {y}])[[1]];gf = (y/.Solve[g == 0, {y}])[[1]];

V = Inverse[A].{t, gf, 1};V = Inverse[A].{t, gf, 1};V = Inverse[A].{t, gf, 1};v = Take[V /V [[3]], 2]v = Take[V /V [[3]], 2]v = Take[V /V [[3]], 2]{−1+t2

t3, −1+t2

t4

}(* gives parameterization *)

Complicated Example of parameterizing curve

P−−−−→

Here P =

[A(x , y)

D(x , y),

B(x , y)

D(x , y)

]where, rounded,

A(x ,y)=0.00409784−2.70587x2+0.830515x+0.587122xy+0.109105y+0.114422y2

B(x ,y)=0.778027+0.585722x2−2.67799x+0.99896xy+0.477164y−0.408743y2

D(x ,y)=−1.80409+1.70004x−0.679318x2+1.84889y+0.706692xy−0.270131y2

Comparision of Graphing Techniques

Consider the intersection of the sphere and ellipsoid

−16 + x2 + y 2 + z2 = 0

14.25− 3x + x2 + y 2/4− 16z + 4z2 = 0

Using the functorial method of graphing this intersection

we get

using two overlapping parameterizations

y 2 = 1.005− 0.1453x + x3

y 2 = 0.1127− 0.0339x + x3

P1−→P2−→

Macaulay and Sylvester Arrays of Polynomials and Ideals

Let f = x − y + 3y3, g = z + 2x2 − 3y2 inC[x , y , z ], F = [f , g , yg ]

The Macaulay Array [?] of F of order 2 at the origin is

1 x y z x2 xy xz y2 yz z2

f 0 1 −1 0 0 0 0 0 0 0g 0 0 0 1 2 0 0 −3 0 0

yg 0 0 0 0 0 0 0 0 1 0

Note that the rows for f , yg are truncated. If we also includedrows for xf , xg , yf , zf , zg we would call this the Macaulay array ofthe basis 〈f , g , 〉.

Macaulay and Sylvester Arrays Continued

I If only rows which correspond to polynomials in F of order n,i.e. not truncated, are included we call the array a SylvesterArray of order n. If all multiples of these rows by monomialswhich still have terms not exceeding total degree n areincluded we have the Sylvester Array of the basis F .

I If we include additional rows so that every polynomial of totaldegree n or less in the ideal generated by the listedpolynomials corresponds to a vector in the rowspace then wewould call this the Sylvester Matrix of the ideal. In theexample of the previous page one would include the rowcorresponding to the polynomial f + yg = x − y + z + 2x2.

I If the Sylvester matrix of a basis F of I is a Sylvester matrixof the ideal generated by the elements of F then F is anH-basis. In other words list F is an H-basis if the Sylvestermatrix of basis F of each order has maximal rank among allSylvester matrices of bases from I.

Global and Local dual spaces as arrays

Local duals can be put in Sylvester type arrays, global in Macaulaytype. We view the dual functionals as columns.

Consider the ideal 〈f 〉 ⊆ C[x , y ] given byf = x + 2y + x2 + 3xy + y2. The local duals are at point (0, 0),indices on right.

Local duals order 2

1 0 0 ∂10 −2 1 ∂x0 1 0 ∂y0 0 4 ∂x20 0 −2 ∂xy0 0 1 ∂y2

Global duals order 2

1 0 0 0 0 10 −2 1 1 0 X0 1 0 0 0 Y0 0 4 −4 −3 X 2

0 0 −2 1 1 XY0 0 1 0 0 Y 2

Note the last two columns of the global duals are truncated.

Local and Global Dual Principle

For large enough order

[Sylvester Matrix

of ideal I

]Macaulayarray ofglobal

duals ofC[x]/I

= 0

[Macaulay array

of ideal I

]Sylvestermatrix of

localduals ofC[x]/I

= 0

In both cases the columns of the right hand matrix span thenullspace of the left hand matrix and the rows of the left handmatrix span the left nullspace of the right hand matrix.

The functorial transformation Gn(φ)

Assume that φ = [f1, . . . , fr ] : As → Ar is a polynomial map whichtakes the origin to the origin. If necessary do a linear translation ofvariables or homogenize. Let X = V (I) be an algebraic set in As

and set J = φ∗−1(I) so that Y = V (J ) = φ(X ).

Construct Gn(φ) as follows:

I For each monomial yk of total degree n or less substituteyi = fi to get gk = f k1

1 · · · f krr ∈ C[x1, . . . , xs ].

I Set Gn(φ) to be the Macaulay matrix of the polynomial list[{gk}]

Theorem: Using large enough n, with high probability

Gn(C[y]/J ) = Gn(φ)Gn(C[x]/I)

Local to Global

For i = [i1, . . . , is ], j = [j1, . . . , js ], i ≥ j means iα ≥ jα for all1 ≤ α ≤ s. Then as functionals on C[x]

∂xj [x] =∑i≥j

(i1j1

)x i1−j11 · · ·

(isjs

)x is−jss Xi

where x = (x1, . . . , xs). The left hand side is a local functional andthe right a global functional. From a matrix point of view we havefor fixed n

Macaulay Matrixof order n

global duals from x

=

Change of Centermatrix

of order n

Sylvester Matrixof order n

local duals at x

Local to Global Theorem

Given an ideal I of C[x1, . . . , xs ], n > 0 and pointspi , i = 1, . . . , k of V (I) concatenate the Macaulay matrices oforder n global duals. Write Dn({p1, . . . , pk}) for this matrix.

Local-Global Theorem, matrix form: For given n > 0 there existfinitely many points, p1, . . . , pk , of V (I) so that the Sylvestermatrix of the ideal I of order n is the left nullspace ofDn({p1, . . . , pk}).

Corollary An H-Basis for I can be obtained from finitely manylocal duals at finitely many points of V (I).

It remains an open question as to how many and what points areneeded. It is clear that it is necessary to have at least one pointfrom each component of V (I) . In principle, for large n, that maybe enough. In practice more points may be needed and the numbermay be dependent on implementation issues as well asalgebraic-geometric factors.

References

B. H. Dayton, Numerical Algebraic Geometry via Numerical PolynomialAlgebra, Talk at AG11, Raleigh NC, October 2011, PDF slides available athttp://www.neiu.edu/∼bhdayton/AG11.pdf

B. H. Dayton, Numerical Calculation of H-bases for PositiveDimensional Varieties, Proceedings of the 4th International Workshop onSymbolic-Numeric Computation (SNC’11). M. Moreno Maza, Editor.ISBN: 978-1-4503-0515-0. ACM Press, 2011. Available athttp://www.csd.uwo.ca/∼moreno//SNC-11-file-for-ACM/snc2011.pdf

B. H. Dayton, The Functorial property of the Global Dual, Talk atMichigan Computational Algebraic Geometry Confrence, May 2012,Oakland University, PDF slides available athttp://www.neiu.edu/∼bhdayton/MCAG12.pdf

B. H. Dayton, T-Y Li, Z. Zeng Multiple Zeros of nonlinear systemsMath. Comp. 80 (2011), no. 276, pp. 2143-2168.

F. S. Macaulay The Algebraic Theory of Modular Systems, CambridgeUniversity Press, 1916.