Symmetric Determinantal Representations Research Articles

Characterizing principal minors of symmetric matrices via determinantal multiaffine polynomials

Here we consider the image of the principal minor map of symmetric matrices over an arbitrary unique factorization domain R. By exploiting a connection with symmetric determinantal representations, we characterize the image of the principal minor map through the condition that certain polynomials coming from so-called Rayleigh differences are squares in the polynomial ring over R. In almost all cases, one can characterize the image of the principal minor map using the orbit of Cayley's hyperdeterminant under the action of (SL2(R))n⋊Sn. Over C, this recovers a characterization of Oeding from 2011, and over R, the orbit of a single additional quadratic inequality suffices to cut out the image. Applications to other symmetric determinantal representations are also discussed.

Hyperbolic secant varieties of M-curves

Abstract We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. For elliptic normal curves, we further obtain definite symmetric determinantal representations for the hyperbolic secant hypersurfaces, which implies the existence of symmetric Ulrich sheaves of rank one on these hypersurfaces. We also use this to derive better bounds on the size of semidefinite representations for convex hulls of real algebraic curves of genus 1.

The 4 × 4 Orthostochastic Variety

ABSTRACT Orthostochastic matrices are the entrywise squares of orthogonal matrices, and naturally arise in various contexts, including notably definite symmetric determinantal representations of real polynomials. However, defining equations for the real variety were previously known only for 3 × 3 matrices. We study the real variety of 4 × 4 orthostochastic matrices, and find a minimal defining set of equations consisting of 6 quintics and 3 octics. The techniques used here involve a wide range of both symbolic and computational methods, in computer algebra and numerical algebraic geometry.

A short proof of the symmetric determinantal representation of polynomials

We provide a short proof of the theorem that every real multivariate polynomial has a symmetric determinantal representation, which was first proved in Helton et al. (2006) [4]. We then provide an example using our approach and extend our results from the real field R to an arbitrary field F different from characteristic 2. The new approach we take is only based on elementary results from the theory of determinants, the theory of Schur complements, and basic properties of polynomials.

Computing symmetric determinantal representations

We introduce the DeterminantalRepresentations package for Macaulay2, which computes definite symmetric determinantal representations of real polynomials. We focus on quadrics and plane curves of low degree (i.e. cubics and quartics). Our algorithms are geared towards speed and robustness, employing linear algebra and numerical algebraic geometry, without genericity assumptions on the polynomials.

The local-global property for bitangents of plane quartics

We study the arithmetic of bitangents of smooth quartics over global fields. With the aid of computer algebra systems and using Elsenhans--Jahnel's results on the inverse Galois problem for bitangents, we show that, over any global field of characteristic different from $2$, there exist smooth quartics which have bitangents over every local field, but do not have bitangents over the global field. We give an algorithm to find such quartics explicitly, and give an example over $\mathbb{Q}$. We also discuss a similar problem concerning symmetric determinantal representations. This paper is a summary of the first author's talk at the JSIAM JANT workshop on algorithmic number theory in March 2019. Details will appear elsewhere.

Definite determinantal representations of multivariate polynomials

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial. Determinantal polynomials can characterize the feasible sets of semidefinite programming (SDP) problems that motivates us to deal with this problem. We introduce the notion of generalized mixed discriminant (GMD) of matrices which translates the determinantal representation problem into computing a point of a real variety of a specified ideal. We develop an algorithm to determine such a determinantal representation of a bivariate polynomial of degree [Formula: see text]. Then we propose a heuristic method to obtain a monic symmetric determinantal representation (MSDR) of a multivariate polynomial of degree [Formula: see text].

A complete characterization of determinantal quadratic polynomials

The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems. In this paper we provide a necessary and sufficient condition for the existence of monic Hermitian determinantal representation as well as monic symmetric determinantal representation of size 2 for a given quadratic polynomial. Further we propose a method to construct such a monic determinantal representation (MDR) of size 2 if it exists. It is known that a quadratic polynomial f(x)=xTAx+bTx+1 has a symmetric MDR of size n+1 if A is negative semidefinite. We prove that if a quadratic polynomial f(x) with A which is not negative semidefinite has an MDR of size greater than 2, then it has an MDR of size 2 too. Finally, we characterize all quadratic polynomials that exhibit MDRs of any size.

Plane quartics: the universal matrix of bitangents

Aronhold’s classical result states that a plane quartic can be recovered by the configuration of any Aronhold systems of bitangents, i.e., special 7-tuples of bitangents such that the six points at which any sub-triple of bitangents touches the quartic do not lie on the same conic in the projective plane. Lehavi (cf. [L05]) proved that a smooth plane quartic can be explicitly reconstructed from its 28 bitangents; this result improved Aronhold’s method of recovering the curve. In a 2011 paper [PSV11] Plaumann, Sturmfels and Vinzant introduced an 8 × 8 symmetric matrix that parametrizes the bitangents of a nonsingular plane quartic. The starting point of their construction is Hesse’s result for which every smooth quartic curve has exactly 36 equivalence classes of linear symmetric determinantal representations. In this paper we tackle the inverse problem, i.e., the construction of the bitangent matrix starting from the 28 bitangents of the plane quartic, and we provide a Sage script intended for computing the bitangent matrix of a given curve.

Determinantal representations of hyperbolic curves via polynomial homotopy continuation

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to compute such representations numerically. Our method works by lifting paths from the space of hyperbolic polynomials to a branched cover in the space of pairs of symmetric matrices.

The local–global principle for symmetric determinantal representations of smooth plane curves in characteristic two

We give an application of Mumford's theory of canonical theta characteristics to a Diophantine problem in characteristic two. We prove that a smooth plane curve over a global field of characteristic two is defined by the determinant of a symmetric matrix with entries in linear forms in three variables if and only if such a symmetric determinantal representation exists everywhere locally. It is a special feature in characteristic two because analogous results are not true in other characteristics.

Computing the determinantal representations of hyperbolic forms

The numerical range of an n × n matrix is determined by an n degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an n degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus g = 1. We reformulate the Fiedler-Helton-Vinnikov formulae for the genus g = 0, 1, and present an elementary computation of the reformulation. Several examples are provided for computing the real symmetric matrices using the reformulation.

On the symmetric determinantal representations of the Fermat curves of prime degree

We prove that the defining equations of the Fermat curves of prime degree cannot be written as the determinant of symmetric matrices with entries in linear forms in three variables with rational coefficients. In the proof, we use a relation between symmetric matrices with entries in linear forms and non-effective theta characteristics on smooth plane curves. We also use some results of Gross–Rohrlich on the rational torsion points on the Jacobian varieties of the Fermat curves of prime degree.

The local–global principle for symmetric determinantal representations of smooth plane curves

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi-Brauer varieties and mod 2 Galois representations, and prove that the local-global principle holds for conics and cubics. We also construct counterexamples to the local-global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.

Symmetric Determinantal Representations in characteristic 2

This paper studies Symmetric Determinantal Representations (SDR) in characteristic 2, that is the representation of a multivariate polynomial P by a symmetric matrix M such that P=det(M), and where each entry of M is either a constant or a variable.We first give some sufficient conditions for a polynomial to have an SDR. We then give a non-trivial necessary condition, which implies that some polynomials have no SDR, answering a question of Grenet et al.A large part of the paper is then devoted to the case of multilinear polynomials. We prove that the existence of an SDR for a multilinear polynomial is equivalent to the existence of a factorization of the polynomial in certain quotient rings. We develop some algorithms to test the factorizability in these rings and use them to find SDRs when they exist. Altogether, this gives us polynomial-time algorithms to factorize the polynomials in the quotient rings and to build SDRs. We conclude by describing the case of Alternating Determinantal Representations in any characteristic.

Determinantal representations of hyperbolic plane curves: An elementary approach

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 able to give a new proof that relies only on the basic intersection theory of plane curves. We show that a matrix of linear forms is definite if and only if its co-maximal minors interlace its determinant and extend a classical construction of determinantal representations of Dixon from 1902. Like the Helton–Vinnikov theorem, this implies that every hyperbolic region in the plane is defined by a linear matrix inequality.

Symmetric determinantal representation of polynomials

We give an elementary proof, only using linear algebra, of a result due to Helton, Mccullough and Vinnikov, which says that any polynomial over the reals can be written as the determinant of a symmetric affine linear pencil. We give explicit determinantal representation formulas and extend our results to polynomials with coefficients in a ring of characteristic different from 2.

Noncommutative convexity arises from linear matrix inequalities

This paper concerns polynomials in g noncommutative variables x=(x1,…,xg), inverses of such polynomials, and more generally noncommutative “rational expressions” with real coefficients which are formally symmetric and “analytic near 0.” The focus is on rational expressions r=r(x) which are “matrix convex” near 0; i.e., those rational expressions r for which there is an ϵ>0 such that if X=(X1,…,Xg) is a g-tuple of n×n symmetric matrices satisfying ϵIn−(X12+⋯+Xg2)is positive definite and Y is also, then the symmetric matrix tr(X)+(1−t)r(Y)−r(tX+(1−t)Y)is positive semidefinite for all numbers t, 0⩽t⩽1. This article gives a complete classification of matrix convex rational expressions (see Theorem 3.3) by representing such r in terms of a symmetric “linear pencil” Lγ(x):=Id−∑jAjxj+(0d−100−1+γ−r(0)) in the noncommuting variables xj, where Aj are symmetric d×d matrices. Namely, for γ a real number, γ−r is a Schur complement of the linear pencil Lγ. Moreover, given a matrix convex r, the set consisting of g tuples X of n×n symmetric matrices (0.1){X:r(X)−γIis negative definite} has component containing 0 which is the same as the “negativity set,” (0.2){X:Lγ(X)is negative definite} for Lγ. Conditions like Lγ(X) is negative definite are known as linear matrix inequalities (LMIs) in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of LMI techniques. In this language what we have shown in (0.1) vs. (0.2) is that the set of solutions to a “convex matrix inequality” with noncommutative unknowns is the same as the set of solutions to some LMI. In many engineering systems problems convexity would have all of the advantages of LMIs. Indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics. Since LMIs have a structure which is seemingly much more rigid than convexity, there is the continual hope that a convexity based theory will be more far reaching than LMIs. But will it? There are two natural situations: one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication. These latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are “dimensionless” in the sense that they scale “automatically with dimension” (as do most of the classics of control theory). That is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than LMIs. Indeed the result proves this for a class of model problems. Furthermore, we show that existing algorithms together with algorithms described here construct the LMIs above which are equivalent to the matrix inequalities based on the given matrix convex rational function r. In a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation, namely, there are symmetric matrices A0,…,Ag in SRd×d with A0 invertible such that (0.3)detp(X)=det(A0−LA(X)) for each X a g-tuple of symmetric n×n matrices. Of course taking n=1 implies immediately that a (commuting variables) polynomial p on Rg has a symmetric determinantal representation. For g=2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher-dimensional case; on the other hand, a nonsymmetric commutative determinantal representation for any g is due to Valiant (“universality of determinant” in algebraic complexity theory). Our determinantal representation theorem is a bi-product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper. While the notion of noncommutative rational functions is standard, the equivalence relation we use on rational expressions in our construction, based on evaluating rational expressions on matrices, is new and gives a new approach to noncommutative rational functions.

Linear symmetric determinantal hypersurfaces

The question which equations of hypersurfaces in the complex projective space can be expressed as the determinant of a matrix whose entries are linear forms is classical. In 1844 Hesse proved that a smooth plane cubic has three essentially different linear symmetric representations [He]. Dixon showed in 1904 that for smooth plane curves linear symmetric determinantal representations correspond to ineffective theta–characteristics, i.e., ineffective divisor classes whose double is the canonical divisor [Di]. Barth proved the corresponding statement for singular plane curves [B]. The general case for any hypersurface was treated by Catanese [C], Meyer–Brandis [M–B], and Beauville [Be].