Sparse resultants: initial forms, vanishing coefficients, homogeneities and generalized Macaulay formulas

CARLOS D'ANDREA; GABRIELA JERONIMO; MARTÍN SOMBRA

Trento

Conferencia; Effective Methods in Algebraic Geometry (MEGA 2015); 2015

We extend previous results and algorithms to sparse resultants associated to arbitrary collections of supports. Among them, we present a factorization formula for the initial part of a sparse resultant which generalizes a previous result by Sturmfels valid only for essentialfamilies of supports under some general hypotheses.We then apply the obtained factorization formula for initial parts to get a crite-rion for the vanishing of sparse resultant after setting some coecients to zero, and afactorization formula for this specialization in the case it does not vanish. This gener-alizes previous results by Minimair, who considered the vanishing of the coecients of only one of the input Laurent polynomials.We also make explicit some degrees of homogenization of sparse resultants in termsof mixed integrals.