Georgia Institute of Technology College of Sciences

A globally nonnegative polynomial F is called stubborn if no odd power of F is a sum of squares. We develop a new invariant of a singularity of a form (homogeneous polynomial) in 3 variables, which allows us to conclude that if the sum of these invariants over all zeroes of a nonnegative form is large enough, then the form is stubborn. As a consequence, we prove that if an extreme ray of the cone of nonnegative ternary sextics is not a sum of squares, then all of its odd powers are also not sums of squares, and we provide more examples of this phenomenon for ternary forms in higher degree. This is joint work with Khazhgali Kozhasov and Bruce Reznick.

Given a basis for a linear subspace U of nxn matrices, we study the problem of either producing a rank-one matrix in U, or certifying that none exist. While this problem is NP-Hard in the worst case, we present a polynomial time algorithm to solve this problem in the generic setting under mild conditions on the dimension of U. Our algorithm is based on Hilbert’s Nullstellensatz and a “lifted” adaptation of the simultaneous diagonalization algorithm for tensor decompositions. We extend our results to the more general setting in which the set of rank-one matrices is replaced by an algebraic set. Time permitting, we will discuss applications to quantum separability testing and tensor decompositions. This talk is based on joint work with Harm Derksen, Nathaniel Johnston, and Aravindan Vijayaraghavan.

Independent component analysis (ICA) is a classical data analysis method to study mixtures of independent sources. An ICA model is said to be identifiable if the mixing can be recovered uniquely. Identifiability is known to hold if and only if at most one of the sources is Gaussian, provided the number of sources is at most the number of observations. In this talk, I will discuss our work to generalize the identifiability of ICA to the overcomplete setting, where the number of sources can exceed the number of observations.The underlying problem is algebraic and the proof studies linear spaces of rank one symmetric matrices. Based on joint work with Anna Seigal https://arxiv.org/abs/2401.14709

A general real rational plane curve C of degree d has 3(d-2) flexes and (d-1)(d-2)/2 complex double points. Those double points lying in RP^2 are either nodes or solitary points. The Welschinger sign of C is (-1)^s, where s is the number of solitary points. When all flexes of C are real, its parameterization comes from a point on the Grassmannian under the Wronskii map, and every parameterized curve with those flexes is real (this is the Mukhin-Tarasov-Varchenko Theorem). Thus to C we may associate the local degree of the Wronskii map, which is also 1 or -1. My talk will discuss work with Brazelton and McKean towards a possible conjecture that these two signs associated to C agree, and the challenges to gathering evidence for this.