Hilbert's Nullstellensatz, Sparse Polynomials and Degree Bounds

HERRERO, MARÍA ISABEL; JERONIMO, GABRIELA; SABIA, JUAN

Ciudad de México

Congreso; XXIII Coloquio Latinoamericano de Álgebra; 2019

Hilbert's Nullstellensatz states that a family F of polynomials has no common zeroes in an algebraically closed field if and only if 1 is in the ideal F generates. In this talk, we present new upper bounds for the degrees of the polynomials appearing in the writing of 1 as a linear combination of the polynomials in F in terms of mixed volumes of convex sets associated with the supports of F. Moreover, for an ideal I, we give an upper bound for the Noether exponent of I (a notion extremely related to Hilbert's Nullstellensatz) also in terms of mixed volumes of convex sets associated with the supports of a family of generators of I. Our bounds are the first one to distinguish the different supports of the arbitrary polynomials involved and can be considerably smaller than previously known ones.