Algorithms for computing syzygies over V[X 1,..., X n ] where V is a valuation ring

Transcription

1 Algorithms for computing syzygies over V[X 1,..., X n ] where V is a valuation ring Ihsen Yengui Department of Mathematics, University of Sfax, Tunisia Graz, 07/07/2016 1

2 Plan Computing syzygies over V[X 1,..., X n ] with Gröbner bases Computing syzygies over V[X 1,..., X n ] via saturation, general case 2

3 a 1,..., a n R. The syzygy module of (a 1,..., a n ) is Syz(a 1,..., a n ) := {(b 1,..., b n ) R n b 1 a b n a n = 0}. A ring V is called a valuation ring if all its elements are comparable under division. A valuation ring is coherent if the annihilator Ann(x) = Syz(x) of any element x V is finitely-generated. 3

4 Definitions 2. Let V be a coherent valuation ring, f, g V[X 1,..., X n ] \ {0}, I = f 1,..., f s a nonzero finitely generated ideal of V[X 1,..., X n ], and > a monomial order. (i) If mdeg(f) = α and mdeg(g) = β then let γ = (γ 1,..., γ n ), where γ i = max(α i, β i ) for each i. The S-polynomial of f and g is the combination: S(f, g) = Xγ LM(f) f LC(f) LC(g) divides LC(f). X γ X γ LM(g) g if LC(g) S(f, g) = LC(g) LC(f) LM(f) f Xγ g if LC(f) LM(g) divides LC(g) and LC(g) does not divide LC(f). (ii) The auto-s-polynomial of f is S(f, f) := d f, where d is a generator of the annihilator of LC(f) (it is defined up to a unit). (iii) G = {f 1,..., f s } is said to be a Gröbner basis for I if LT(I) = LT(f 1 ),..., LT(f s ). 4

5 Theorem 2. Let V be a coherent valuation ring, I = g 1,..., g s an ideal of V[X 1,..., X n ], and fix a monomial order >. Then, G = {g 1,..., g s } is a Gröbner basis for I if and only if for all pairs 1 i j s, the remainder on division of S(g i, g j ) by G is zero. Buchberger s Algorithm for Coherent valuation rings Input: g 1,..., g s V[X 1,..., X n ], V a coherent valuation ring, > a monomial order Output: a Gröbner basis G for g 1,..., g s with {g 1,..., g s } G G := {g 1,..., g s } REPEAT G := G For each pair f, g in G DO S := S(f, g) G If S 0 THEN G := G {S} UNTIL G = G 5

6 Example: Let V[X] = (Z/16Z)[X], and consider the ideal I = f 1, where f 1 = 2+4X+8X 2. S(f 1, f 1 ) = 2f 1 = 4 + 8X =: f 2, S(f 1, f 2 ) = 2 =: f 3, S(f 2, f 2 ) = 2f 2 = 8 f 3 0, S(f 3, f 3 ) = 0, f 2 f 3 0. Thus, G = {2} is a Gröbner basis for I in V[X]. 6

7 Theorem. Let V be a valuation ring. Then, one can construct Gröbner bases over V (for the lexicographic monomial order) if and only if V is both coherent and archimedean (i.e., a, b Rad(V) \ {0} n N a divides b n ), or also, if and only if either dim V 1 and V is without zero-divisors or dim V = 0 and the annihilator of any element in V is finitely generated. 7

8 References: Yengui I. Dynamical Gröbner bases. J. Algebra 301 (2006) Hadj Kacem A., Yengui I. Dynamical Gröbner bases over Dedekind rings. J. Algebra 324 (2010) Lombardi H., Schuster P., Yengui I. The Gröbner ring conjecture in one variable. Math. Z. 270 (2012) Yengui I. The Gröbner Ring Conjecture in the lexicographic order case. Math. Z. 276 (2014)

9 It is a folklore that if V is valuation domain then V[X 1,..., X n ] is coherent, i.e., Syzygy modules over V[X 1,..., X n ] are finitely generated. This follows from a deep and complicated paper: Gruson L., Raynaud M. Critères de platitude et de projectivité. Invent. Math. (1971). Our goal is to find an algorithm for computing syzygies over V[X 1,..., X n ], where V is a valuation domain of any Krull dimension. Let p 1,..., p m V[X 1,..., X k ], and consider n vectors s 1,..., s n V[X 1,..., X k ] m generating the syzygy module of p 1,..., p m over the quotient field K of V as a V[X 1,..., X k ]-module (s 1,..., s n can be computed using Gröbner bases techniques). Then, the syzygy module S of p 1,..., p m over V is nothing but the V-saturation of S = s 1,..., s n, i.e., S := {s V[X 1,..., X k ] m α s S for some α V \ {0}} = (S R K) V[X 1,..., X k ] m. 9

10 V = Z 2Z, s 1 = (5, 4, 2X 2 6X + 12) height (0, 1) index 6 n = 5 d = 4 d + 1 = deg (PrimMon) [defect=2] 10

11 V = Z 2Z ; S := [s 1 = (5, 4, 2X 2 6X + 12), s 2 = (2X 1, 0, 2X 2 + 6X 4)] reduction S 0 = [(5, 4, 2X 2 6X + 12), (X, 2 5, 6 5 X X 4 5 )]; δ(s 0) = 1 XS 0 reduction S 1 = [(5X, 4X, 2X 3 6X X), (0, 2X 1, X 3 + 2)]; δ(s 1 ) = 0 As a conclusion Sat(s 1, s 2 ) = (5, 4, 2X 2 6X+12), (X, 2 5, 6 5 X X 4 5 ), (0, 2X 1, X 3 + 2). 11

12 Theorem: Let S = [s 1,..., s n ] be a finite list of vectors in V[X] m with degrees d, where V is a valuation domain and m 1. Then the primitive triangulation algorithm computes after min(n 1, m)d+1 iterations a finite list G of vectors in V[X] m of degrees (min(n 1, m) + 1)d generating Sat( s 1,..., s n ) as a V[X]-module. In other terms, computing Sat( s 1,..., s n ) amounts to performing gaussian elimination on a matrix of size n(min(n 1, m)d+1) m and with entries in V[X] of degrees (min(n 1, m)+1)d. Proof. We denote by S 0 the list S put in an echelon form, and by induction T j = [S 0,..., S j ] where S j+1 denotes XS j put in an echelon form with respect to T j and then put in an echelon form, with the initialization T 0 = S 0. Then the sequence (δ(s j )) j 0 is non-increasing and becomes zero for j min(n 1, m)d. 12

13 Theorem: Let L be a finite list of vectors in V[X 1,..., X k ] m, where V is a valuation domain of quotient field K and residue field k. Then dim k L dim K L, L V is V-saturated if and only if dim K L = dim k L. 13

14 When a matrix over the integers is Z- saturated? A Z m n ; rk 0 A := rk Q A; rk p A := rk Fp A; P = the set of prime numbers; P := P {0}. Denoe by p 1,..., p t the prime numbers dividing the denominators of the vectors obtained after putting the columns of A into an echelon form over Q. Then the following assertions are equivalent: (i) Im(A) is Z-saturated. (ii) rk 0 A = rk p1 A = = rk pt A. (iii) The map rk(a) : P N defined by rk(a)(q) := rk q A, is constant. (iv) The map P N; p rk p A, is constant. 14

15 Let L = [u 1,..., u s ] (s 1) be a list of s polynomial vectors in V[X 1,..., X k ] m, where V is a valuation domain of quotient field K and residue field k. For i N, L i := Mu j ; 1 j s, tdeg(m) i K, L i := Mū j ; 1 j s, tdeg(m) i k. h L, K (t) = i 0(dim K L i ) t i, h L, k (t) = i 0(dim k L i ) t i h L, K (t), δ L (t) := h L, K (t) h L, k (t) called the saturation defect series of the list L. Note that h L,K (t) = HS SyzK (u 1,...,u s ) (t). 15

16 Example: Consider the list U = [u 1 = 1 + 2X, u 2 = 1 + 2Y ] with u i Z 2Z [X, Y ]. We have: h U, Q (t) = 1 (1 t) (1 t) 2, h U, Z/2Z (t) = 1 (1 t) 3, and, thus, the defect series of U is δ U (t) = 1 (1 t) 2. 16

17 Theorem: Let L be a finite list of vectors in V[X 1,..., X k ] m, where V is a valuation domain. If δ L = 0 then L V[X1,...,X k ] is V-saturated. 17

18 Saturation algorithm in the multivariate case: Input: A finite list S = [s 1,..., s n ] of vectors in V[X 1,..., X k ] m, where V is a valuation domain and m 1. Output: A finite list G of vectors in V[X 1,..., X k ] m generating Sat( s 1,..., s n ) as a V[X 1,..., X k ]-module. We denote by S 0 the list S put in an echelon form, and by induction T j = [S 0,..., S j ] where S j+1 denotes [X 1 S j,..., X k S j ] put in an echelon form with respect to T j and then put in an echelon form, with the initialization T 0 = S 0. We begin by putting S in an echelon form (it becomes S 0 ) and then compute its defect series δ S0 (t). If δ S0 (t) = 0 then stop; else compute S 1. If δ S1 (t) = 0 then stop; else compute S 2, and so on. 18

19 Example: U = [u 1 = 1 + 2X, u 2 = 1 + 2Y ] with u i Z 2Z [X, Y ]. As δ U (t) = 1 (1 t) 2 0, one has to put U in an echelon form. This can be done as follows: U = [u 1 = 1 + 2X, u 2 = 1 + 2Y ] U 0 := [u 1, 1 2 (u 2 u 1 )] = [1 + 2X, Y X]. As h U0, Q(t) = h U0, Z/2Z (t) = 1 have δ U0 (t) = 0. We conclude that (1 t) (1 t) 2, we Sat( u 1, u 2 ) = 1 + 2X, Y X. 19

20 References: Lombardi H., Quitté C., Yengui I. Un algorithme pour le calcul des syzygies sur V[X] dans le cas où V est un domaine de valuation. Communications in Algebra 42:9 (2014) Ducos L., Monceur S., Yengui I. Computing the V-saturation of finitely-generated submodules of V[X] m where V is a valuation domain. J. Symb. Comput., in press. Ducos L., Valibouze A., Yengui I. Computing syzygies over V[X 1,..., X k ], V a valuation domain. J. Algebra 425 (2015) Yengui I. Constructive Commutative Algebra. Lecture Notes in Mathematics, no 2138, Springer

21 DANKE 21