A single exponential time algorithm for homogeneous regular sequence tests

Abstract

Assume that we are given a sequence [Formula: see text] of [Formula: see text] homogeneous polynomials in [Formula: see text] variables of degree at most [Formula: see text] and the ideal [Formula: see text] generated by this sequence. The aim of this paper is to present a new and effective method to determine, within the arithmetic complexity [Formula: see text], whether [Formula: see text] is regular. This algorithm has been implemented in Maple and its efficiency (compared to the classical approaches for regular sequence test) is evaluated via a set of benchmark polynomials. Furthermore, we show that if [Formula: see text] is regular then we can transform [Formula: see text] into Nœther position and at the same time compute a reduced Gröbner basis for the transformed ideal within the arithmetic complexity [Formula: see text]. Finally, under the same assumption, we establish the new upper bound [Formula: see text] for the maximum degree of the elements of any reduced Gröbner basis of [Formula: see text] in the case that [Formula: see text].