Decomposable sparse polynomial systems

Random Sparse Polynomial Systems

Abstract Let f :=(f1, . . . , fn) be a sparse random polynomial system. This means that each f i has fixed support (list of possibly non-zero coefficients) and each coefficient has a Gaussian probability distribution of arbitrary variance. We express the expected number of roots of f inside a region U as the integral over U of a certain mixed volume form. When U = (C∗)n, the classical mixed vol...

Counting Stable Solutions of Sparse Polynomial Systems

Global Residues for Sparse Polynomial Systems

We consider families of sparse Laurent polynomials f1, . . . , fn with a finite set of common zeroes Zf in the torus T n = (C − {0}) . The global residue assigns to every Laurent polynomial g the sum of its Grothendieck residues over Zf . We present a new symbolic algorithm for computing the global residue as a rational function of the coefficients of the fi when the Newton polytopes of the fi ...

Extremal Sparse Polynomial Systems over Local Fields

Consider a system F of n polynomials in n variables, with a total of n + k distinct exponent vectors, over any local field L. We discuss conjecturally tight upper and lower bounds on the maximal number of non-degenerate roots F can have over L, with all coordinates having fixed sign or fixed first digit, as a function of n and k only. In particular, for non-Archimedean L, we give the first non-...

Affine solution sets of sparse polynomial systems

This paper focuses on the equidimensional decomposition of affine varieties defined by sparse polynomial systems. For generic systems with fixed supports, we give combinatorial conditions for the existence of positive dimensional components which characterize the equidimensional decomposition of the associated affine variety. This result is applied to design an equidimensional decomposition alg...