Kristian Ranestad

The algebraic degree of semidefinite programming

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.

Algebraic Degree of Polynomial Optimization

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, we prove that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials. We also give a general formula for the algebraic degree of the optimal coordinates. The derivation of the algebraic degree is equivalent to counting the number of all complex critical points. As special cases, we obtain the algebraic degrees of quadratically constrained quadratic programming (QCQP), second order cone programming (SOCP), and

On the cactus rank of cubic forms

p

A general formula for the algebraic degree in semidefinite programming

th order cone programming (POCP), in analogy to the algebraic degree of semidefinite programming [J. Nie, K. Ranestad, and B. Sturmfels, The algebraic degree of semidefinite programming, Math. Programm., to appear].

Algebraic boundaries of Hilbert’s SOS cones

We prove that the smallest degree of an apolar 0-dimensional scheme of a general cubic form in n+1 variables is at most 2n+2, when n>=8, and therefore smaller than the rank of the form. For the general reducible cubic form the smallest degree of an apolar subscheme is n+2, while the rank is at least 2n.

THE ABELIAN FIBRATION ON THE HILBERT CUBE OF A K3 SURFACE OF GENUS 9

In this note, we use a natural desingularization of the conormal variety of the variety of n x n symmetric matrices of rank at most r to find a general formula for the algebraic degree in semidefinite programming.

The Convex Hull of a Variety

We study the geometry underlying the difference between non-negative polynomials and sums of squares (SOS). The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether–Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized, respectively, by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

Syzygies of Abelian and Bielliptic Surfaces in ℙ4

In this paper we construct an abelian fibration over

THE GEOMETRY OF BIELLIPTIC SURFACES IN ℙ4

{\bf P}^3

Quartic spectrahedra

on the Hilbert cube of the primitive K3 surface of genus 9. After the abelian fibration constructed by Mukai on the Hilbert square on the primitive K3 surface S of genus 5, this is the second example where the abelian fibration on such Hilb_n(S) is directly constructed. Our example is also the first known abelian fibration on a Hilbert scheme Hilb_n(S) of a primitive K3 surface S which is not the Hilbert square of S; the primitive K3 surfaces on the Hilbert square of which such a fibration exists are known by a recent result of Hassett and Tschinkel.