Various algorithms for the computation of Bernstein-Sato polynomial

Various algorithms for the computation of Bernstein-Sato polynomial Applications of Computer Algebra (ACA) 2008

Notations and Definitions Motivation Two Approaches Let K be a field and let D = K x, = K x 1,..., x n, 1,..., n {x i j = j x i + δ ij } denote the n-th Weyl algebra over K. Definition (Initial Form) Let 0 w R n be a weight vector. For p = α,β c αβx α β D put m = max α,β { wα + wβ c αβ 0}. We call in ( w,w) (p) := c αβ x α β D α,β: wα+wβ=m the initial form of p w.r.t. to w.

Notations and Definitions Motivation Two Approaches Let K be a field and let D = K x, = K x 1,..., x n, 1,..., n {x i j = j x i + δ ij } denote the n-th Weyl algebra over K. Definition (Initial Form) Let 0 w R n be a weight vector. For p = α,β c αβx α β D put m = max α,β { wα + wβ c αβ 0}. We call in ( w,w) (p) := c αβ x α β D α,β: wα+wβ=m the initial form of p w.r.t. to w. For an ideal I in D, in ( w,w) (I ) := K {in ( w,w) (p) p I } is called the initial ideal of I w.r.t. w.

Notations and Definitions Motivation Two Approaches Definitions (Global b-function) Let I be a holonomic D-ideal and 0 w R n. Set s := n i=1 w ix i i. Then in ( w,w) (I ) K[s] is a non-zero principal ideal in K[s]. We call its monic generator b(s) the global b-function of I w.r.t. w.

Notations and Definitions Motivation Two Approaches Definitions (Global b-function) Let I be a holonomic D-ideal and 0 w R n. Set s := n i=1 w ix i i. Then in ( w,w) (I ) K[s] is a non-zero principal ideal in K[s]. We call its monic generator b(s) the global b-function of I w.r.t. w. For f K[x] put I f := t f, i + f x i t i = 1,..., n D t, t. Let w = (1, 0,..., 0) R n+1 be a weight vector such that the weight of t is 1, and let B(s) be the global b-function of I f w.r.t. w. We call b(s) := B( s 1) the global b-function (or the Bernstein-Sato polynomial) of f.

Notations and Definitions Motivation Two Approaches Some applications of the b-function The b-function, respectively its minimal/maximal integer roots, are needed in other areas of D-module theory: in computing restrictions in computing integrations in computing localizations in computing operators and annihilators

Notations and Definitions Motivation Two Approaches Some applications of the b-function The b-function, respectively its minimal/maximal integer roots, are needed in other areas of D-module theory: in computing restrictions in computing integrations in computing localizations in computing operators and annihilators: b(s) is the uniquely determined monic polynomial satisfying the identity P(x,, s) f s+1 = b(s) f s D[s]/ Ann(f s ) for some P(x,, s) D[s].

Notations and Definitions Motivation Two Approaches Annihilator based methods The idea: The equation implies P(x,, s) f s+1 = b(s) f s P(x,, s) f b(s) Ann(f s ).

Notations and Definitions Motivation Two Approaches Annihilator based methods The idea: The equation implies P(x,, s) f s+1 = b(s) f s P(x,, s) f b(s) Ann(f s ). 1 Compute Ann(f s ) (e.g. by using the algorithms by Oaku-Takayama, Levandovskyy-Oaku-Takayama or Briançon-Maisonobe) 2 Compute b(s) = Ann(f s ) D[s], f K[s] (e.g. by using preimages) Cf. the talk of Viktor Levandovskyy and Jorge Morales.

Notations and Definitions Motivation Two Approaches Initial based methods The idea: Back to the roots

Notations and Definitions Motivation Two Approaches Initial based methods The idea: Back to the roots 1 Compute J = in ( w,w) (I ) 2 Compute b(s) = J K[s] for s = w i x i i

Notations and Definitions Motivation Two Approaches Initial based methods The idea: Back to the roots 1 Compute J = in ( w,w) (I ) 2 Compute b(s) = J K[s] for s = w i x i i Oaku, Takayama: 1 Homogenize and compute a Gröbner basis G of J 2 Compute J = J K[x 1 1,..., x n n ] by Gröbner basis elimination, compute J K[s] by commutative methods

Computation of in ( w,w) (I ) Computation of the initial ideal Computation of the intersection Lemma Let < be a term order and let < ( w,w) be the (non-term) monomial order defined by x α β < ( w,w) x γ δ wα + wβ < wγ + wδ or wα + wβ = wγ + wδ and x α β < x γ δ. If G is a Gröbner basis of I w.r.t. < ( w,w), then G ( w,w) = {in ( w,w) (g) g G} is a Gröbner basis of in ( w,w) (I ) w.r.t. <.

Weighted homogenization Computation of the initial ideal Computation of the intersection Let u, v R n >0. Consider D(h) (u,v) = K x,, h with non commutative relations i x i = x i i + h u i +v i, 1 i n. For p = c λαβ h λ x α β define the weighted total degree of p: deg (u,v) (p) = max{λ + uα + vβ c λαβ 0} For p = c αβ x α β define the weighted homogenization of p: H (u,v) (p) = c αβ h deg (u,v) (p) (uα+vβ) x α β

Computation of the initial ideal Computation of the intersection For a monomial order < in D define a term order < h in D (h) (u,v) : p < h q deg (u,v) (p) < deg (u,v) (q) or deg (u,v) (p) = deg (u,v) (q) and p h=1 < q h=1 Theorem Let F be a non-empty subset of D. If G h is a Gröbner basis of H (u,v) (F ) w.r.t. < h, then G h h=1 is a Gröbner basis of F w.r.t. <.

Computation of in ( w,w) (I ) K[s] Computation of the initial ideal Computation of the intersection The idea: Interpret s = n i=1 w ix i i as a left D-endomorphism D/ in ( w,w) (I ) D/ in ( w,w) (I ), p p s.

Computation of in ( w,w) (I ) K[s] Computation of the initial ideal Computation of the intersection The idea: Interpret s = n i=1 w ix i i as a left D-endomorphism D/ in ( w,w) (I ) D/ in ( w,w) (I ), p p s. Well definition: For every p in ( w,w) (I ), p s = (s + m) p in ( w,w) (I ), where m = wα + wβ for any non-zero term c α,β x α β in p.

Computation of in ( w,w) (I ) K[s] Computation of the initial ideal Computation of the intersection The idea: Interpret s = n i=1 w ix i i as a left D-endomorphism D/ in ( w,w) (I ) D/ in ( w,w) (I ), p p s. Well definition: For every p in ( w,w) (I ), p s = (s + m) p in ( w,w) (I ), where m = wα + wβ for any non-zero term c α,β x α β in p. Known fact: For a holonomic D-ideal J, Hom D (D/J, D/J) is a finite dimensional K-vector space. Hence, s has a well defined minimal polynomial.

Computation of the initial ideal Computation of the intersection Algorithm: MinimalPolynomial(s, G) Input: s D such that G K[s] {0} for a GB G of J Output: b(s) K[s] such that J K[s] = b(s) i 1 while 1 do if there exist a 0,..., a i 1 K such that NF(s i, G) + i 1 j=0 a j NF(s j, G) = 0 then return s i + i 1 j=0 a js j break else i i + 1 end if end while

A modular alteration Introduction Computation of the initial ideal Computation of the intersection Consider K = Q. For a prime p let Z (p) = { a b a, b Z, b / pz} and let φ p : Z (p) Z p be the canonical projection. Lemma (Noro) For a prime p such that G Z (p) [x, ], φ p (G) is a Gröbner basis of φ p (G) w.r.t. < and φ p (b(s)) φ p (G). Theorem (Noro) Let b p (s) be the minimal polynomial of φ p (s) in φ p (D)/ φ p (G). If there exists f Z[s] such that deg(f (s)) = deg(b p (s)) and f (s) G, then f (s) = b(s).

The implementation To appear: bfct.lib http://www.singular.uni-kl.de/

Example Input uw 1 xyz(y z)(y + z) uw 2 xyz(x + y + z)(y z) uw 3 xyz(x + z)(y z) uw 4 xyz(x + y + z)(3x + 2y + z) uw 5 xyz(y z)(2y + z)(y + z) uw 11 xyz(x z)( x + y)(y + z) uw 12 xyz(x z)( x + y)(y z) uw 13 xyz(4x + 2y + z)(9x + 3y + z)(x + y + z) uw 14 xyz(2y + z)(y + z)(4y + z)(3y + z) uw 15 xyz( x + y z)(3y + z)(2y + z)(y + z) uw 16 xyz(x z)(2y + z)(3y + z)(y + z) uw 19 xyz( x + y + 2z)(x + y + 2z)(y z)(y + z) uw 20 xyz(x z)(x + z)(y z)(y + z) uw 22 xyz(x + z)( x + y)(y z)(y + z) uw 32 xyz(x z)(x y z)(x y)(y z)

Asir Singular Example deg(b(s)) bfct bernsteinbm bfct uw 1 7 0:01 0:01 0:01 uw 2 10 10:52 0:15 0:07 uw 3 9 0:04 0:02 0:03 uw 4 8 1h:09 0:40 0:04 uw 5 9 0:01 0:01 0:05 uw 11 11 2h:51 13h:29 0:22 uw 12 11 17:00 42h:46 0:19 uw 13 10 37h:42 n/a 0:22 uw 14 11 0:04 0:03 0:35 uw 15 18 27h:07 n/a 32:55 uw 16 19 23h:55 n/a 13:28 uw 19 18 33h:44 n/a 2:13 uw 20 15 06h:47 55h:51 0:33 uw 22 17 28h:43 n/a 3:25 uw 32 13 24h:24 n/a 2:52 t : out of memory after time t, t : killed by user after time t

Ongoing Work bfct is not yet able to compute the b-function of (xz + y)(x 4 + y 5 + xy 4 ) Hyperplane arrangements in more variables Yet unknown: the b-function of xyzw(x + y)(x + z)(x + w)(y + z)(y + w) Other types of polynomials Yet unknown: the b-function of (z 3 + w 4 )(3z 2 x + 4w 3 y) Applications in theoretical physics: kind of ζ-functions

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