I will discuss the problem of estimating the volume of the tube around an algebraic set (possibly singular) as a function of the dimension of the set and its degree. In particular I will show bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any smoothness assumption. This problem is related to numerical algebraic geometry, where sets of ill-contidioned inputs are typically described by (possibly singular) algebraic sets, and their neighborhoods describe bad-conditioned inputs. Our result generalizes previous work of Lotz on smooth complete intersections in the euclidean space and of Bürgisser, Cucker and Lotz on hypersurfaces in the sphere, and gives a complete solution to Problem 17 in the book titled "Condition" by Bürgisser and Cucker. This is joint work with S. Basu.

Landau in 1905 proved that every univariate polynomial with rational coefficients which is strictly positive on the reals is a sum of squares of rational polynomials. However, it is still not known whether univariate rational polynomials which are non-negative on all the reals, rather than strictly positive, are sums of squares of rational polynomials.

In this talk we consider the local counterpart of this problem, namely, we consider rational polynomials that are non-negative on the real roots of another non-zero rational polynomial. Parrilo in 2003 gave a simple construction that implies that if \(f\) in \(\mathbb R[x]\) is squarefree and \(g\) in \(\mathbb R[x]\) is non-negative on the real roots of \(f\) then \(g\) is a sum of squares of real polynomials modulo \(f\). Here, inspired by this construction, we prove that if \(g\) is a univariate rational polynomial which is non-negative on the real roots of a rational polynomial \(f\) (with some condition on \(f\) wrt \(g\) which includes squarefree polynomials) then it is a sum of squares of rational polynomials modulo \(f\).

Joint work with Bernard Mourrain (INRIA, Sophia Antipolis) and Agnes Szanto (North Carolina State University).

The number of roots of random polynomials have been intensively studied for a long time. In the case of systems of polynomial equations the first important results can be traced back to the nineties when Kostlan, Shub and Smale computed the expectation of the number of roots of some random polynomial systems with invariant distributions. Nowadays this is a very active field.

In this talk we are concerned with the variance and the asymptotic distribution of the number of roots of invariant polynomial systems. As particular examples we consider Kostlan-Shub-Smale, random spherical harmonics and Real Fubini Study systems.

Real symmetric matrices of size n, whose all principal submatrices of size \(k \lt n\) are positive semidefinite, form a closed convex cone. Such matrices do not need to be PSD and, in particular, they can have negative eigenvalues. The geometry of the set of eigenvalues of all such matrices is far from being completely understood. In this talk I will show that already when \((n,k)=(4,2)\) the set of eigenvalues is not convex.

We study the conditioning of algebraic computational problems. We have particular interest in which known theorems about conditioning on particular problems can be extended to the general context. We will pay special attention to the case in which the set of inputs with a common output is not a linear subspace.

In my previous talk at CLAM-2021, I stated a complexity bound for solving sparse systems of polynomial equations in terms of condition numbers, mixed volume and surface and a few technical details. This results uses a technique that I call renormalization, and the bound excludes systems with solution at toric infinity.

In a recent paper, Duff, Telen, Walker and Yahl proposed a homotopy algorithm using Cox coordinates. This representation associates one new coordinate to each ray (one-dimensional cone) on the fan of the toric variety.

In this talk I will report on recent progress inspired by their work, on non-degenerate solutions at toric infinity for sparse polynomial systems.