Computing the Bernstein-Sato polynomial

Transcription

1 Computing the Bernstein-Sato polynomial Daniel Andres Kaiserslautern Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

2 Overview 1 Introduction 2 Computing the initial ideal 3 Intersecting an ideal with a subalgebra The univariate case The multivariate case 4 The s-parametric annihilator 5 Wish list Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

3 The Weyl algebra Definition Let K be a field of characteristic 0. The K-algebra D = K x, = K x 1,..., x n, 1,..., n { i x j = x j i + δ ij } is a G-algebra, the n-th Weyl algebra over K. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

4 The initial ideal Definition Let 0 w R n 0. For 0 p = α,β c αβx α β D set Then m := max α,β { wα + wβ c αβ 0}. in ( w,w) (p) := α,β: wα+wβ=m c αβ x α β is called the initial form of p w.r.t. w. Additionally, set in ( w,w) (0) := 0. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

5 The initial ideal Definition Let 0 w R n 0. For 0 p = α,β c αβx α β D set Then m := max α,β { wα + wβ c αβ 0}. in ( w,w) (p) := α,β: wα+wβ=m c αβ x α β is called the initial form of p w.r.t. w. Additionally, set in ( w,w) (0) := 0. For an ideal I D we call the initial ideal of I w.r.t. w. in ( w,w) (I) := K {in ( w,w) (p) p I} Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

6 b-function and Bernstein-Sato polynomial Definition Let I D be an ideal, 0 w R n 0 und s := n i=1 w i x i i. The monic generator b(s) of in ( w,w) (I) K[s] is called the (global) b-function of I w.r.t. w. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

7 b-function and Bernstein-Sato polynomial Definition Let I D be an ideal, 0 w R n 0 und s := n i=1 w i x i i. The monic generator b(s) of in ( w,w) (I) K[s] is called the (global) b-function of I w.r.t. w. Forr f K[x] \ K definine the Malgrange-Ideal I f of f to be I f := t f, i + f x i t i = 1,..., n D t, t. Let w = (1, 0,..., 0) R n+1 such that t gets the weight 1 and let B(s) be the b-function of I f w.r.t. w. Then b(s) := ( 1) deg(b(s)) B( s 1) is called the (global) b-function or the Bernstein-Sato polynomial of f. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

8 Computing the initial ideal Let 0 w R n 0, u, v Rn >0 and let be a global ordering on D. x w w D Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

9 Computing the initial ideal Let 0 w R n 0, u, v Rn >0 and let be a global ordering on D. x x h w w homogenization u v 1 w w 0 D D (h,u,v) := K x,, h { i x j = x j i + δ ij h u j+v i } Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

10 Computing the initial ideal Let 0 w R n 0, u, v Rn >0 and let be a global ordering on D. x x h w w homogenization u v 1 w w 0 =: (h,u,v) ( w,w) D D (h,u,v) := K x,, h { i x j = x j i + δ ij h u } j+v i Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

11 Algorithm (initialidealw) Input: I D ideal, 0 w R n 0, global ordering on D, u, v Rn >0 Output: Gröbner basis G of in ( w,w) (I) w.r.t. G (h) := a Gröbner basis of the homogenization of I w.r.t. (h,u,v) ( w,w) return G = in ( w,w) (G (h) h=1 ) Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

12 Statement of the problem A J A G s A \ K associative K-algebra ideal finite Gröbner basis of J w.r.t. an arbitrary global ordering arbitrary Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

13 Statement of the problem A J A G s A \ K associative K-algebra ideal finite Gröbner basis of J w.r.t. an arbitrary global ordering arbitrary What do we know about J K[s]? Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

14 Statement of the problem A J A G s A \ K associative K-algebra ideal finite Gröbner basis of J w.r.t. an arbitrary global ordering arbitrary What do we know about J K[s]? We distinguish between the following situations: 1 lm(g) lm(s k ) for all g G, k N 0 2 Situation 1 does not apply: 1 J s J and dim K (End A (A/J)) < 2 Situation 2.1 does not apply: 1??? Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

15 Situations 1 and 2 Lemma (Situation 1) If there is no g G with lm(g) lm(s k ) for a k N 0, then J K[s] = {0}. Beweis: Definition of Gröbner basis. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

16 Situations 1 and 2 Lemma (Situation 1) If there is no g G with lm(g) lm(s k ) for a k N 0, then J K[s] = {0}. Beweis: Definition of Gröbner basis. Remark (Situation 2) The converse is not true. Consider J = y 2 + x K[x, y]. Then J K[y] = {0}, but {y 2 + x} is a Gröbner basis of J for any ordering. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

17 Situation 2.1 Lemma (Situation 2.1) Let J s J and End A (A/J) (A/J viewed as A-module) finite dimensional as K-vector space. Then J K[s] {0}. Proof: s : A/J A/J, [a] [a s] is a well-defined A-module endomorphism. s has a minimal polynomial µ. µ K[s] J and µ 0 (even µ / K). Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

18 Situation 2.1 Lemma (Situation 2.1) Let J s J and End A (A/J) (A/J viewed as A-module) finite dimensional as K-vector space. Then J K[s] {0}. Proof: Remark s : A/J A/J, [a] [a s] is a well-defined A-module endomorphism. s has a minimal polynomial µ. µ K[s] J and µ 0 (even µ / K). Especially, the second condition holds, if A/J itself is a finite dimensional A-module, or A is a Weyl algebra and A/J is a holonomic module. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

19 Back to the b-function Theorem The b-function of a holonomic ideal I D is non-constant. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

20 Back to the b-function Theorem The b-function of a holonomic ideal I D is non-constant. Proof: Let 0 w R n 0, J := in ( w,w)(i) for a holonomic ideal I D and s := n i=1 w i x i i. D/J is a holonomic D-module. Endomorphism spaces of holonomic modules are finite dimensional. For ( w, w)-homogeneous p J it holds that p s = (s + m) p J, where m = deg ( w,w) (p). Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

21 Algorithm (pintersect, to be changed to uniintersect) Input: s A \ K, J A such that J K[s] {0} Output: b K[s] monic such that J K[s] = b G := finite Gröbner basis of J w.r.t. an arbitrary global ordering i := 1 loop if there exist a 0,..., a i 1 K satisfying NF(s i, G) + i 1 j=0 a j NF(s j, G) = 0 then return b := s i + i 1 j=0 a js j else i := i + 1 end if end loop Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

22 Tuning of the normal form Lemma Let A be a G-algebra of Lie type (e.g. a Weyl algebra), J A an ideal and s A. For i N put r i = NF(s i, J) and q i = s i r i J. Then it holds for all i N that Proof: r i+1 = NF([s i r i, r 1 ] + r i r 1, J). s i+1 = q i s + r i s = q i (q 1 + r 1 ) + r i (q 1 + r 1 ) = q i q 1 + q i r 1 + r i q 1 + r i r 1 q i r 1 + r i r 1 [q i, r 1 ] + r i r 1 = [s i r i, r 1 ] + r i r 1 Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

23 Applications Computation of b-functions Solving of zerodimensional systems Computation of central characters Example (Computation of central characters) Universal enveloping algebra of the Lie algebra sl 2 : A = U(sl 2, K) = K e, f, h [e, f] = h, [h, e] = 2e, [h, f] = 2f Center of A: Z(A) = K[4ef + h 2 2h] F = {e 11, f 12, h 5 10h 3 + 9h} A L = A F as left ideal, T = A F A as two-sided ideal Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

24 Algorithm (intersectupto, to be changed to multiintersect) Input: s 1,..., s r A \ K pairw. comm., J A ideal, k N Output: Gröbner basis B of J K[s 1,..., s r ] up to degree k G := Gröbner basis of J w.r.t. an arbitrary ordering d := 0, B := while d k do M d := {s α α d} if a m K with a m NF(m, G) = 0 then m M d f := a m m m M d if f 0 and f / B then B := B {f} end if end if d := d + 1 end while return B Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

25 Improvements If p B mit lm(p) = m has been found, ignore all multiples of m in the following steps. If G is a Gröbner basis of J w.r.t., forget about {m M d max (m L(G) M d ) m}, since p J K[s 1,..., s r ] lm(p) L(G) K[s 1,..., s r ]. NF(m, G) = m, if m / L(G) K[s 1,..., s r ]. General stop criterion without degree bound? Only clear, if J K[s 1,..., s r ] is zerodimensional, or known to be a principal ideal. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

26 s-parametric annihilator The ring K[s, x, f 1, f s ] becomes a D K K[s] = D[s]-module via x i g(s, x)f s+j := x i g(s, x)f s+j, s g(s, x)f s+j := s g(s, x)f s+j, i g(s, x)f s+j := g x i f s+j + (s + j)g f x i f s+j 1. Definition The ideal Ann(f s ) = {p D[s] p f s = 0} is called the (s-parametric) annihilator of f s. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

27 Bernstein s Theorem Theorem (Bernstein) The Bernstein-Sato polynomial b(s) of f K[x 1,..., x n ] \ K is the uniquely determined, monic polynomial of smallest degree in K[s], satisfying the identity P f s+1 = b(s) f s for some P D[s]. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

28 Bernstein s Theorem Theorem (Bernstein) The Bernstein-Sato polynomial b(s) of f K[x 1,..., x n ] \ K is the uniquely determined, monic polynomial of smallest degree in K[s], satisfying the identity P f s+1 = b(s) f s for some P D[s]. Thus, we obtain an alternate algorithm to compute b(s): (P f b(s)) f s = 0, i.e. P f b(s) Ann(f s ), hence b(s) f + Ann(f s ). Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

29 Two algorithms Algorithm (bfctann) Input: f K[x 1,..., x n ] \ K Output: The Bernstein-Sato polynomial b(s) K[s] of f G := Gröbner basis of f + Ann(f s ) G D[s] b(s) := pintersect(s, G) s indeterminate return b(s) Algorithm (bfct) Input: f K[x 1,..., x n ] \ K Output: The Bernstein-Sato polynomial b(s) K[s] of f w := (1, 0,..., 0), s := t t G := initialidealw(i f, w) G D t, t B(s) := pintersect(s, G) s polynomial return b(s) = ( 1) deg(b(s)) B( s 1) Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

30 Improvements to bfctann Since ( Ann(f s ) + f + f ) i = 1,..., n K[s] = b(s) x i s + 1, one saves one NF computation. For every it holds that (y 0, y 1,..., y n ) syz K[x] (f, f x 1,..., f x n ) n y 0 s + y i i Ann(f s ), i=1 yielding elements of smallest possible degree in i. Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

31 Singular wish list modular methods for G-algebras (Gröbner basis and intersection) Hilbert-driven Gröbner basis computation in G-algebras Gröbner walk techniques for G-algebras computation of Ann(f s ) in the kernel, making use of the identity Ann(f s ) K[x, s] = {0} in the involved Gröbner basis computation more efficient computation of NF of powers of polynomials generalization of finduni to polynomials and G-algebras fast Gaussian elimination generalization of (parts of) FGLM? Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21

32 Singular wish list modular methods for G-algebras (Gröbner basis and intersection) Hilbert-driven Gröbner basis computation in G-algebras Gröbner walk techniques for G-algebras computation of Ann(f s ) in the kernel, making use of the identity Ann(f s ) K[x, s] = {0} in the involved Gröbner basis computation more efficient computation of NF of powers of polynomials generalization of finduni to polynomials and G-algebras fast Gaussian elimination generalization of (parts of) FGLM? Thank you! Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern / 21