Understanding, finding, or even deciding on the existence of real solutions to a system of
equations is a difficult problem with many applications outside of mathematics. While it is …

For an ideal I⊆ ℝ [x] given by a set of generators, a new semidefinite characterization of its
real radical I (V ℝ (I)) is presented, provided it is zero-dimensional (even if I is not) …

A polynomial identity testing algorithm must determine whether an input polynomial (given
for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a …

This graduate textbook presents an approach through toric geometry to the problem of
estimating the isolated solutions (counted with appropriate multiplicity) of n polynomial …

We give a multivariate version of Descartes' rule of signs to bound the number of positive
real roots of a system of polynomial equations in variables with monomials, in terms of the …

We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture.
Along the way, we illustrate a computer-assisted approach to finding sparse polynomial …

We investigate a version of Viro's method for constructing polynomial systems with many
positive solutions, based on regular triangulations of the Newton polytope of the system. The …

Consider a system f_1(x)=0,...,f_n(x)=0 of n random real polynomial equations in n variables,
where each f_i has a prescribed set of exponent vectors described by a set A⊆N^n of …

We present an optimal version of Descartes' rule of signs to bound the number of positive
real roots of a sparse system of polynomial equations in n variables with n+ 2 monomials …

We show that weakly reversible mass-action systems can have a continuum of positive
steady states, coming from the zeroes of a multivariate polynomial. Moreover, the same is …