We seek to determine a real algebraic variety from a fixed finite subset of points. Existing
methods are studied and new methods are developed. Our focus lies on aspects of topology …

Abstract 10 years ago or so Bill Helton introduced me to some mathematical problems
arising from semidefinite programming. This paper is a partial account of what was and what …

If a real polynomial f can be written as a sum of squares of real polynomials, then clearly f is
nonnegative on Rn, and an explicit expression of f as a sum of squares is a certificate of …

Let A(x)=A_0+x_1A_1+⋯+x_nA_n be a linear matrix, or pencil, generated by given
symmetric matrices A_0,A_1,...,A_n of size m with rational entries. The set of real vectors x …

Landau's work on the singularities of Feynman diagrams suggests that they can only be of
three types: either poles, logarithmic divergences, or the roots of quadratic polynomials. On …

In 2007, Helton and Vinnikov proved that every hyperbolic plane curve has a definite real
symmetric determinantal representation. By allowing for Hermitian matrices instead, we are …

Abstract Recent work of Kass–Wickelgren gives an enriched count of the 27 lines on a
smooth cubic surface over arbitrary fields. Their approach using A 1-enumerative geometry …

Quartic spectrahedra in 3-space form a semialgebraic set of dimension 24. This set is
stratified by the location of the ten nodes of the corresponding real quartic surface. There are …

We study real double covers of branched over a-divisor, which are conic bundles with
smooth quartic discriminant curve by the second projection. In each isotopy class of smooth …

For every bivariate polynomial p (z_1, z_2) p (z 1, z 2) of bidegree (n_1, n_2)(n 1, n 2), with p
(0, 0)= 1 p (0, 0)= 1, which has no zeros in the open unit bidisk, we construct a determinantal …