Computational algebraic geometry

Transcription

1 Computational algebraic geometry Learning coefficients via symbolic and numerical methods Anton Leykin Georgia Tech AIM, Palo Alto, December 2011

2 Let k be a field (R or C). Ideals varieties Ideal in R = k[x 1,..., x n ] generated by f 1,..., f r is Variety defined by I is I = f 1,..., f r = r g i f i R i=1 V = V(I) = {p k n : f I, f(p) = 0} The vanishing ideal of a variety V is I = I(V ) = {f R : f(v ) = 0} The radical of I is I = {g : m, g m I} R, V(I) = V( I) I is a radical ideal if I = I Hilbert Nullstellensatz: if k = C then I(V(J)) = J for all J R.

3 Gröbner bases, invariants A monomial order > is given by weights w 1, w 2,... R n 0 : x α > x β w 1 α > w 1 β or w 2 α > w 2 β or... such that > is total. Example: > lex is given by w 1 = e 1, w 2 = e 2,.... The initial term of f R is the term (coefficient monomial) with the largest monomial occurring with a nonzero coefficient: e.g. for f = 3x 2 y + 4xy it is in lex (f) = 3x 2 y. A Gröbner basis of I is a set of generators G of I such that in(g) = in(i). Invariants such as dimension, degree, Hilbert polynomial can be computed using Gröbner bases.

4 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) Sage, Maple, Mathematica,... (more general software)

5 Polynomial homotopy continuation Target system: n equations in n variables, F (x) = (f 1 (x),..., f n (x)) = 0, where f i R = C[x] = C[x 1,..., x n ] for i = 1,..., n. Start system: n equations in n variables: G(x) = (g 1 (x),..., g n (x)) = 0, such that it is easy to solve. Homotopy: for γ C \ {0} consider H(x, t) = (1 t)g(x) + γtf (x), t [0, 1].

6 Example target start f 1 = x 4 1x 2 + 5x 2 1x x g 1 = f 2 = x 2 1 x 1 x 2 + x 2 8 g 2 = x x Start solutions target solutions: H(x, t) = 0 implies dx ( ) 1 H dt = H x t.

7 Global picture Optimal homotopy: the continuation paths are regular; the homotopy establishes a bijection between the start and target solutions. Possible singular scenarios: non-generic diverging paths multiple solutions

8 Higher-dimensional solution sets Let I = (f 1,..., f N ) be an ideal of C[x 1,..., x n ]. Goal: Understand the variety X = V(I) = {x C n f I, f(x) = 0}. A witness set for an equidimensional component Y of X: a generic slicing plane L with dim L = codim Y witness points w Y,L = Y L (generators of I)

9 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].

10 Analysis For a polynomial f, the function f s is locally integrable Re s > 0. Hence, f s is a generalized function defined on {s : Re s > 0} C: for a distribution ϕ Cc, f s (ϕ) = f(x) s ϕ(x) dx Gelfand [1957]: Does it extend to a meromorphic function on C? I. N. Bernstein [1968]: Yes. The poles are contained in a finite number of arithmetic progressions. Key ingredients: resolution of singularities and being able to write a functional equation b(s)f s = P f s+1 when f is a monomial. (Here: b(s) is a univariate polynomial and P is a linear differential operator with coefficients in C[x, s].)

11 Invariants in singularity theory Definition (Multiplier ideal for f = (f 1,..., f r )) J (f c ) = { h 2 } h C[x] : ( f i 2 is locally integrable. ) c For r = 1, it is the ideal of h, that make h f 1 c locally integrable. Algebrao-geometric definition: via log-canonical resolutions. Jumping coefficients of f: rational numbers 0 = ξ 0 < ξ 1 < ξ 2 < such that J (f c ) is constant exactly for c [ξ i, ξ i+1 ). ξ 1 is called the log-canonical threshold. These invariants measure singularities of the corresponding variety; in particular, they depend only on the ideal f.

12 Weyl algebra Let K be a field of characteristic zero. (Think: K = C) Affine space: X = K n. Weyl algebra: an associative algebra D X = K x, = K x 1,..., x n, 1,..., n where [ i, x i ] = i x i x i i = 1 and all other pairs of generators commute. D X is isomorphic to the algebra of linear differential operators with polynomial coefficients. Every element has the normal form Q = c αβ x α β, α,β Z n where finitely many of c αβ K are nonzero.

13 D-modules D X is simple: only trivial two-sided ideals. We consider only left ideals and left D X -modules. Examples of D-modules: K[x], K[[x]], C (X). Software: kan/sm1 (Takayama) risa/asir (Noro) dmod.lib, Singular (Levandovsky et al.) D-modules, Macaulay2 (L., Tsai)

14 Gröbner bases D X is Gröbner-friendly: D X is an algebra of solvable type. Gröbner bases can be computed with respect to any w-compatible monomial order, where w = (w x, w ) R 2n satisfies w x + w 0 componentwise. The Bernstein-Sato polynomial b f (s) 0 is the monic polynomial b(s) of the minimal degree satisfying b(s)f s = P f s+1 = P f f s D X [s]f s, where P D X [s] and D X [s]f s is a cyclic D X [s]-module generated by f s. One can computed b f (s) via Gröbner bases.

15 Connection between lct(f) and b f (s) Assume k = C then The log-canonical threshold c 0 is the lowest root of b f ( s). Every jumping coefficient c [c 0, c 0 + 1) is a root of b f ( s). For k = R there are examples where rlct(f) is not a jumping coefficient of f. However, rlct(f) mod 1 equals some root of b f ( s). M2 packages where lct is computed: Dmodules (general case) MonomialMultiplierIdeals (monomial ideals) HyperplaneArrangements (central hyperplane arrangements)

16 Improper integrals To find c = rlct(f) look for the pole s = c closest to 0 of f s (ϕ) = f(x) s ϕ(x) dx. x R n Idea: Pick ϕ supported 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) = then c 1 < rlct(f) c 2. We can determine possible values of rlct(f) by looking at b f (s)?

17 Questions Can we say something about the denominator of rlct(f)? How to compute an improper integral numerically? How to determine it equals? Can we use statistics to estimate rlct? Is there a hybrid strategy (e.g., partial resolution + b-functions)?