Upper and lower bounds for the degree of Groebner bases

Abstract

The problem of the complexity of Buchberger's algorithm to compute GrSbner bases has been recently studied by Buchberger [2,3],Baye~ l],and Lazar~ 7].Here we present some results on this question,by giving both lower and upper bounds for the maximal degree of the elements of a Gr6bner basis of a polynomial ideal,as a function of the degree of a general basis,the number of variables and the dimension of the ideal.To know the complexity of Buchberger's algorithm,the knowledge of a bound for the degree of the GrSbner basis is not sufficient:also a bound for the degree of the polynomials arising during the intermediate computations is required. Unfortunately we are able to obtain such a bound only under assumptions on the input basis (namely,an Hbasis of the ideal must be known). Our examples show that this condition is unavoidable.ii) the second class of examples is more related with a kind of basis (H-basis) which has weaker properties than a Gr6bner basis and which is widely studied in classical constructive ideal theory.ln her well-known pape~ 61 ,Hermann gave an incorrect proof that the maximal degree of an H-basis is doubly exponential in the number of variables.Here we adapt an example by Mayr and Meyer[8] to show the existence of ideals such that any H-basis (and henceforth any Gr~bner basis under a degree compatible term-ordering)contains elements whose degree is doubly exponential in the dimension of the ideal.As for upper bounds,Bayer [l] proved that the degree of a Gr6bner basis is bounded by a coefficient (under a suitable representation) of the Hilbert polynomial of the ideal (the use of Hilbert polynomials to give a bound to the degree of a Gr6bner basis has

Reduction Mod p of Standard Bases

“…It would be interesting to obtain an explicit (say doubly-exponential) bound β, as for Gröbner bases in Dubé (1990), Möller and Mora (1984). The inductive argument used in the proof of Proposition 3.14 also shows:…”

Complexity of a standard basis of a $D$-module

“…2 O(n) and that the absolute values of all coefficients of the Hilbert polynomial of I are bounded from above by (dl) 2 O(n) ; cf., e.g., [12]. This fact follows directly from (11), Lemma 12 in Appendix 1, Lemma 2, and Theorem 2.…”

“…In the present paper we fill this very essential gap and prove a double-exponential upper bound for complexity. On the other hand, a double-exponential complexity lower bound for Gröbner bases [12,15] provides by the same token a bound for Janet bases.…”

“…The Janet bases generalize the Gröbner bases, which were widely used in the algebra of polynomials (see, e.g., [3]). For the Gröbner bases, a double-exponential complexity bound was obtained in [12,6] with the help of [1]. Later, sharper results on the same subject (with independent and self-contained proofs) were obtained in [4].…”

“…Notice also that there has been a folklore opinion that the problem of constructing a Janet basis reduces easily to the commutative case by considering the associated graded module, and, on the other hand, in the commutative case [6,12,4], the doubleexponential upper bound is well known.…”