Abstract
Gröbner basis methods are used to solve systems of polynomial equations over finite fields, but their complexity is poorly understood. In this work an upper bound on the time complexity of constructing a Gröbner basis according to a total degree monomial ordering and finding a solution of a system is proved. A key parameter in this estimate is the degree of regularity of the leading forms of the polynomials. Therefore, we provide an upper bound to the degree of regularity for a sufficiently overdetermined system of forms of the same degree over any finite field. The bound holds for almost all polynomial system and depends only on the number of variables, the number of polynomials, and the degree. Our results imply that almost all sufficiently overdetermined systems of polynomial equations of the same degree are solvable in polynomial time.
Highlights
Let x1, . . . , xn be variables over a field
Are a main object to study in algebraic geometry and commutative algebra
Buchberger [4] defined the notion of a Gröbner basis for a polynomial ideal and showed how to construct such a basis
Summary
Are a main object to study in algebraic geometry and commutative algebra. Several methods to find an explicit solution to (1) were developed. Lazard [16] showed that a Gröbner basis according to a total degree monomial ordering may be constructed by triangulating a suitable Macaulay matrix. Among equation systems of the same degree, those which are overdetermined may be solved faster than those where m ≤ n using algorithms from Gröbner basis of XL families [2,6]. In Theorem 2.1 we show that the time-complexity of constructing a total degree Gröbner basis for (1) is polynomial in Lq(n, dreg), where Lq(n, d) is the number of monomials in Rh of total degree at most d. 2.1 and 2.2, a total degree Gröbner basis and a solution to a relevant equation system may be computed in polynomial time. An extended abstract of this paper was presented at WCC2019 [19]
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