A criterion for detectingm-regularity

Abstract

Let S=k[x~ ..... x,] be a polynomial ring over an infinite field k, and let 1 be a homogeneous ideal of S.An algorithm for computing the (first) syzygies of I is due independently to Spear [Spe77] and Schreyer [Sch80]: One chooses an ordering on the monomials of S, and then constructs a monomial ideal in(1) generated by the lead terms of all elements of I, in(l) can be viewed as the limit of 1 under the action of a 1-parameter subgroup of GL(n) on the Hilbert scheme [Bay 82], so in(1) occurs as the special fiber of a flat family whose general fiber is isomorphic to I. It follows from a well-known criterion for flatness [Art 76] that each syzygy of in(l) can be lifted to a syzygy of I; the set of syzygies thus obtained can be trimmed to give a complete set of minimal syzygies of I.The rnonomial ideal in(I) was first studied The following problem arises in using this syzygy algorithm in practice: in(l) can have minimal generators and syzygies in degrees higher than any minimal generator or syzygy of I. In this situation, computations in these higher degrees are unnecessary; one should compute the generators and syzygies of in(1) in only those degrees necessary to find all minimal syzygies of 1.In order to modify the syzygy algorithm to take advantage of this observation, one would like a criterion for determining when all minimal syzygies of I have been found. This problem appears to be intractable at present. However, the question of bounding the degrees of the minimal jth syzygies of I, for all j, is tractable. Recall that I is defined to be m-regular if the jth syzygy module of I is generated in degrees <m+j, for all j>0 ([Mum66], [EiGo 84]).The regularity of I, reg(l), is defined to be the least m for which I is m-regular. We have reg(in (1) We give in w a criterion for 1 to be m-regular, which depends only on computations in the finite vector spaces S,, and S"+I of polynomials of degrees m, m+l:If one can find h 1 .... ,hieS 1, so that the subspaces ((I, h 1 ..... hi-1) : hi)m and (I, hi, ..., h i 1),. are equal for 1 s j, and (I, h 1 .... , hj)m=Sm, then I is m-regular. Furthermore, if I is m-regular, then a generic choice of h~, ..., hje S 1 will satisfy these conditions.One could use this result to terminate syzygy computations early, in cases where reg(in(I))>reg(I). However, further study reveals a close connection between this result and a particular order on the monomials of S, the reverse lexicographic order. This order is used to compute saturations in [Bay 82]. The reverse lexicographic order was then studied in characteristic zero, in generic coordinates, by several authors. It is observed in [Laz83] that under this hypothesis in low dimensions, the generators of in(I) are of particularly low degree. In [Giu84], this hypothesis is further studied, and a worst-case upper bound on the degrees of generators of in(l) is obtained, which improves Hermann's corresponding bound for ideal membership [Her 26]. In [Ang84], independent of a preliminary version of our results, it is shown that under this hypoth...

Some Remarks on Borel Type Ideals

“…. , x n ], n ≥ 2 the polynomial ring over K. Bayer and Stillman [2] note that Borel fixed ideals I ⊂ S satisfy the following property: ( * ) (I : x ∞ j ) = (I : (x 1 , . .…”

“…Combining these facts, we are done. (1) I is of Borel type, (2) For any monomial u ∈ I and for any integer 1 ≤ j < m(u), there exists an integer…”

Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations

“…Gröbner bases are also highly sensitive to the monomial order arranging the different monomials of a multivariate polynomial (see, e.g., [6, Chapter 2] for formal definitions). The Degree Reverse Lexicographic (degrevlex) order gives (on average) Gröbner bases with the smallest total degree [2], although there exist known examples (cf. Mora's example in [14]) for which, even for the degrevlex monomial ordering, the (reduced) Gröbner basis contains a polynomial of total degree O(d 2 ).…”

Rigid resolutions and big Betti numbers

“…By the Bayer-Stillman theorem, cf. [5] or [10] we have reg Gin(I j ) = j, too. Now we apply a result of Eisenbud, Reeves and Totaro [11,Proposition 10].…”