Computational algebraic geometry

Computational algebraic geometry

Learning coefficients via symbolic and numerical methods

Anton Leykin

Georgia Tech

Ideals

↔

varieties

• Letkbe a field (RorC). • IdealinR=k[x1, . . . , xn]generated byf1, . . . , fris I=hf1, . . . , fri= r X i=1 gifi⊂R • Varietydefined byIis V =_V(I) ={p∈kn:∀f ∈I, f(p) = 0}

• Thevanishing idealof a varietyV is

I=_I(V) ={f ∈R:f(V) = 0} • TheradicalofIis √ I={g:∃m, gm∈I} ⊂R, V(I) =V( √ I) Iis a radical ideal ifI=√I

• Hilbert Nullstellensatz: ifk=CthenI(V(J)) =

√

Gröbner bases, invariants

• Amonomial order>is given by weightsw1, w2, . . .∈Rn≥0:

xα> xβ ⇔ w1·α > w1·β or

w2·α > w2·β or

. . . such that>is total.

Example: >lex is given byw1=e1, w2=e2, . . ..

• The initial term off ∈Ris the term (coefficient·monomial) with the largest monomial occurring with a nonzero coefficient:

e.g. forf = 3x2_{y+ 4xy}7_{+ 2it isin}

lex(f) = 3x2y. • AGröbner basisofIis a set of generatorsGofIsuch that

hin(G)i=hin(I)i.

• Invariantssuch as dimension, degree, Hilbert polynomial can be computed using Gröbner bases.

Computational (symbolic) algebraic geometry

• Cox, Little, O’Shea,Ideals, varieties, and algorithms

• Cox, Little, O’Shea,Using algebraic geometry

• Greuel, Pfister,A Singular Introduction to Commutative Algebra

Software:

• Singular(can resolve singularities) • Macaulay2(can do everything else)

• Magma(Australia),CoCoA(Italy),risa/asir(Japan), ... (other specialized software)

Polynomial homotopy continuation

• Targetsystem: nequations innvariables, F(x) = (f1(x), . . . , fn(x)) =0, wherefi∈R=C[x] =C[x1, ..., xn]fori= 1, ..., n. • Startsystem:nequations innvariables:

G(x) = (g1(x), . . . , gn(x)) =0,

such that it is easy to solve.

• Homotopy: forγ∈C\ {0}consider

Example

target start

f1 = x41x2+ 5x21x23+x31−4 g1 = x51−1

f2 = x21−x1x2+x2−8 g2 = x22−1

Start solutions→target solutions: H(x, t) = 0implies dx dt =− ∂H ∂x −1 ∂H ∂t .

Global picture

Optimal homotopy:

• the continuation paths areregular; • the homotopy establishes a bijection

between the start and target solutions. Possiblesingularscenarios:

Higher-dimensional solution sets

• LetI= (f1, . . . , fN)be an ideal ofC[x1, . . . , xn]. • Goal: Understand the variety

X=V(I) ={x∈Cn| ∀f ∈I, f(x) = 0}. • Awitness setfor an equidimensional componentY ofX:

• a generic “slicing” planeLwithdimL= codimY

• witness pointswY,L=Y ∩L • (generators ofI)

Numerical algebraic geometry

• Sommese, Verschelde, and Wampler,Introduction to Numerical AG(2005)

• Sommese and Wampler,The numerical solution of systems of polynomials(2005)

Software:

• PHCpack(Verschelde); • HOM4PS(group of T.Y.Li); • Bertini(group of Sommese);

• NAG4M2: Numerical Algebraic Geometry for Macaulay2 (L.). and more, e.g.:Maple’s ROOTFINDING[HOMOTOPY].

Analysis

• For a polynomialf, the function|f|s_{is locally integrableRes >0.} • Hence,|f|s_{is a generalized function defined on}

{s: Res >0} ⊂C: for a distributionϕ∈Cc∞,

|f|s_{(ϕ) =} Z

|f(x)|s_ϕ(x)dx

• Gelfand [1957]: Does it extend to a meromorphic function onC? • I. N. Bernstein [1968]: Yes. The poles are contained in a finite

number of arithmetic progressions.

• Key ingredients:resolution of singularitiesand being able to write afunctional equation

b(s)fs=P·fs+1

whenf is a monomial. (Here:b(s)is a univariate polynomial and P is a linear differential operator with coefficients inC[x, s].)

Invariants in singularity theory

Definition (Multiplier ideal for

f

= (f1

, . . . , f

r

)

)

J(fc) = h∈C[x] : |h|2 (P |fi|2₎c is locally integrable . Forr= 1, it is the ideal ofh, that make _|f|h|

1|c locally integrable.

• Algebrao-geometric definition: via log-canonical resolutions. • Jumping coefficients off: rational numbers

0 =ξ0< ξ1< ξ2<· · ·

such thatJ(fc)is constant exactly forc∈[ξi, ξi+1).

• ξ1is called thelog-canonical threshold.

• These invariants measure singularities of the corresponding variety; in particular, they depend only on the idealhfi.

Weyl algebra

• LetKbe a field of characteristic zero. (Think:K=C) • Affine space: X =Kn_.

• Weyl algebra: an associative algebra

DX =Khx,∂i=Khx1, . . . , xn, ∂1, . . . , ∂ni

where[∂i, xi] =∂ixi−xi∂i= 1and all other pairs of generators commute.

• DXis isomorphic to the algebra oflinear differential operators

with polynomial coefficients. • Every element has thenormal form

Q= X

α,β∈Zn

cαβxα∂β,

D

-modules

• DXis simple: only trivial two-sided ideals.

• We consider onlyleftideals andleftDX-modules. • Examples ofD-modules:K[x],K[[x]],C∞(X). • Software:

• kan/sm1(Takayama)

• risa/asir(Noro)

• dmod.lib, Singular(Levandovsky et al.)

Gröbner bases

• DXis Gröbner-friendly: DX is an algebra ofsolvable type. • Gröbner bases can be computed with respect to any

w-compatible monomial order, wherew= (wx, w∂)∈R2n satisfieswx+w∂ ≥0componentwise.

• TheBernstein-Satopolynomialbf(s)6= 0is the monic polynomial b(s)of the minimal degree satisfying

b(s)fs=P·fs+1=P f·fs∈DX[s]fs, whereP ∈DX[s]andDX[s]fsis a cyclicDX[s]-module generated byfs_.

Connection between

lct(

f

)

and

b

f

(

s

)

• Assumek=Cthen

• The log-canonical thresholdc0is the lowest root ofbf(−s). • Every jumping coefficientc∈[c0, c0+ 1)is a root ofbf(−s). • Fork=Rthere are examples whererlct(f)is not a jumping

coefficient off.

However,rlct(f) mod 1equals some root ofbf(−s).

• M2 packages wherelctis computed: • Dmodules(general case)

• MonomialMultiplierIdeals(monomial ideals)

Improper integrals

To findc= rlct(f)look for the poles=−cclosest to 0 of

|f|s(ϕ) = Z

x∈Rn

|f(x)|sϕ(x)dx. Idea:

• Pickϕsupported on a neighborhood of a singularity of the variety{f = 0}and evaluate|f|s_{(ϕ)numerically.}

• If|f|−c1_{(ϕ)<∞and|f|}−c2_{(ϕ) =∞thenc}

1<rlct(f)≤c2.

Questions

• Can we say something about the denominator ofrlct(f)? • How to compute an improper integral numerically?

How to determine it equals∞? • Can we use statistics to estimaterlct?