A Sparse Effective Nullstellensatz

Abstract

We present bounds for the sparseness in the Nullstellensatz. These bounds can give a much sharper characterization than degree bounds of the monomial structure of the polynomials in the Nullstellensatz in case that the input system is sparse. As a consequence we derive a degree bound which can substantially improve the known ones in case of a sparse system.In addition we introduce the notion of algebraic degree associated to a polynomial system of equations. We obtain a new degree bound which is sharper than the known ones when this parameter is small. We also improve the previous effective Nullstellensatze in case the input polynomials are quadratic.Our approach is completely algebraic, and the obtained results are independent of the characteristic of the base field. ᮊ 1999 Academic Press

Sparse effective membership problems via residue currents

“…Now ae ≤ deg Φ so that (6.2) is in particular satisfied if Vol(P) ≤ ad µ−1 /µ. Thus Theorem 1.5 improves Kollár's result if the volume of the Newton polytope of the F j is small compared to ad µ−1 /µ, see also [33]. An analogous analysis shows that Theorems 1.3 and 1.4 improve the results by Andersson-Götmark and Hickel, respectively, if Vol(P) ≤ ad µ−1 .…”

“…The restriction d = 2 was removed by Jelonek, [21], for m ≤ n. For m ≥ n + 1 Sombra, [33], proved that one can find G j that satisfy deg (F j G j ) ≤ (1 + degΦ)2 n+1 .…”

“…The main ingredient in Sombra'a proof is an effective Nullstellensatz for arithmetically Cohen-Macaulay varieties, [33,Lemma 1.1]. This result was later extended to general varieties by Kollár,[23], EinLazarsfeld, [17], and Jelonek, [21].…”

Sharp estimates for the arithmetic Nullstellensatz

“…For d = 2 , Sombra [50] recently showed that the bound deg g i f i ≤ 2 n+1 holds. Now, let us consider the height aspect: assume f 1 , .…”

“…[34]). The first general sparse Nullstellensatz was obtained by Sombra [50]. In both cases the authors give bounds for the Newton polytopes of the output polynomials in terms of the Newton polytopes of the input ones.…”