arXiv subjects

Aljaž Zalar

Publications and source records attributed to Aljaž Zalar.

At least 19 recordsLinked to original sources

arXiv2026

Truncated Moment Problems and the Extension Property on Monomial Curves

In several papers, Stochel and Szafraniec studied moment problems on algebraic sets from an operator-theoretic perspective, investigating when positive definite sequences satisfying polynomial relations admit representing measures. Within this framework, Stochel introduced type A sets, and Bisgaard classified the plane curves defined by relations between two monomials that have this property. Curto and Fialkow introduced a stronger, truncated version of the type A property, requiring that the existence of a positive semidefinite extension of prescribed degree guarantees the existence of a representing measure. Motivated by Bisgaard's classification, we determine which plane curves defined by relations between two monomials satisfy this extension property. In the affirmative cases, we obtain explicit bounds on the required extension degree. In the negative cases, we construct truncated sequences that admit positive semidefinite extensions of arbitrarily high order but have no representing measure supported on the curve. These constructions yield explicit polynomials that are nonnegative on the corresponding curves but are not sums of squares in their coordinate rings. In the affirmative cases, we also derive explicit degree bounds for sums-of-squares certificates of strictly positive polynomials.

math.FA↗

arXiv2026

Unobstructedness of affine Gorenstein terminal toric fourfolds

We prove that every affine terminal Gorenstein toric variety of dimension at most four is unobstructed. In dimension four, the obstruction space need not vanish; instead, we determine the possible homogeneous obstruction degrees and construct simultaneous two-parameter deformations which eliminate the remaining potential obstructions.

math.AG↗

arXiv2026

Cyclic polynomials in Dirichlet-type Spaces of the unit bidisk

For $α\in \mathbb{R},$ we consider the scale of function spaces, namely the Dirichlet-type space ${D}_α$ consisting of holomorphic functions on the unit bidisk $\mathbb{D}^2$, $f(z,w)=\sum_{k,l=0}^{\infty}a_{kl}z^kw^l$ such that $$\sum_{k,l=0}^{\infty}(k+l+1)^α|a_{kl}|^2 < \infty.$$ In this paper, we solve an open problem posed by Torkinejad Ziarati concerning the cyclicity of the polynomial $2-z_1-z_2$ in $ D_α$ for $ \frac32 < α\leq 2$. We provide an affirmative answer and, as a consequence, complete the characterization of cyclic polynomials in $ D_α$.

math.FA↗

arXiv2026

A constructive approach to the truncated moment problem on cubic curves in Weierstrass form

In this paper, we develop a constructive solution for the pure truncated moment problem on cubic curves in Weierstrass form, establishing the existence of a representing measure whose number of atoms equals the rank of the associated moment matrix. By a recent result of Baldi, Blekherman, and Sinn, for projectively smooth curves whose projective closure has exactly one real point at infinity, the existence of such a rank-attaining atomic measure is equivalent to the existence of a representing measure; consequently, the TMP is constructively solved for this class of curves. We also present a numerical degree--$6$ example in which every minimal representing measure supported on the cubic curve requires $\operatorname{rank} M(3)+1$ atoms, where $M(3)$ denotes the moment matrix. Finally, we provide a constructive solution for the symmetric case, i.e., when all moments of odd degree in $y$ vanish.

math.FA↗

arXiv2026

Maximal signed volume for (multivariate) supermodular quasi-copulas

Copulas are the primary tool for dependence modeling in statistics, and quasi-copulas are their essential companions. The latter appear, say, as infima or suprema of sets of copulas; they form a huge class and have some unpleasant properties. Their statistical interpretation is challenged by the fact that they may lead to negative volumes of some boxes. So, numerous applications call for an intermediate class, and supermodular quasi-copulas are one of them, having many useful properties. An excellent measure, Average Rectangular Volume (ARV in short), to clarify and position this class was proposed in the seminal paper by Anzilli and Durante, The average rectangular volume induced by supermodular aggregation functions, J. Math. Anal. Appl. 555 (2026) 21 pp. While supermodularity is a bivariate notion, its extension to the $d$-variate case for $d>2$ was recently emphasized in a key paper by Arias-Garcia, Mesiar, and De Baets, The unwalked path between quasi-copulas and copulas: Stepping stones in higher dimensions, Int. J. of Appr. Reasoning, 80 (2017) pp. 89-99. Here, an alternative method to ARV is presented, extendable to the multivariate case based on Maximal (in absolute value) Negative Volumes (MNV in short) on boxes, thus helping practitioners when seeking the right (quasi-)copula for their problem. Observe that these volumes on copulas are zero, while their values on quasi-copulas, depending on $d$, have been a long-standing open problem solved only recently. We present a nontrivial extension of this solution, which serves as the main goal of this paper: a measure that clarifies and positions the classes considered based on MNV.

math.ST↗

arXiv2026

Non-negative polynomials on generalized elliptic curves

We study the cone of non-negative polynomials on generalized elliptic curves. We show that the zero set of every extreme ray has dense real points. If a generalized elliptic curve is embedded via a complete linear system, then we show that the convex hull of its real points (taken inside any affine chart containing all real points) is a spectrahedron. On the way, we generalize a result by Geyer--Martens on 2-torsion points in the Picard group of smooth real curves (of arbitrary genus) to possibly singular and reducible ones.

math.FA↗

arXiv2026

Matrix Fejér-Riesz type theorem for a union of an interval and a point

The matrix Fejér-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. In the previous work of the second-named author this was extended to the characterization on arbitrary closed semialgebraic sets $K$ in $\mathbb{R}$ by using matrix quadratic modules from real algebraic geometry. In the compact case there is a denominator-free characterization, while in the non-compact case denominators are needed except when $K$ is the whole line, an unbounded interval, a union of two unbounded intervals, and it was conjectured also when $K$ is a union of an unbounded interval and a point or a union of two unbounded intervals and a point. In this paper, we confirm this conjecture by solving the truncated matrix-valued moment problem (TMMP) on a union of a bounded interval and a point. The presented technique for solving the corresponding TMMP can potentially be used to determine degree bounds in the positivity certificates for matrix polynomials on compact sets $K$.

math.FA↗

arXiv2025

Extreme mass distributions for quasi-copulas

A recent survey, nicknamed "Hitchhiker's Guide", J.J. Arias-Garcıa, R. Mesiar, and B. De Baets, A hitchhiker's guide to quasi-copulas, Fuzzy Sets and Systems 393 (2020) 1-28, has raised the rating of quasi-copula problems in the dependence modeling community in spite of the lack of statistical interpretation of quasi-copulas. In our previous work (Fuzzy Sets and Systems 517 (2025) 109457), we addressed the question of extreme values of the mass distribution associated with multivariate quasi-copulas. Using a linear programming approach, we were able to solve Open Problem 5 of the "Guide" up to dimension d = 17 and disprove a recent conjecture on the solution to that problem. In this paper, we use an analytical approach to provide a complete answer to the original question.

stat.ML↗

arXiv2025

The pure $Y=X^{d}$ truncated moment problem

Let $β\equivβ^{(2n)}$ be a real bivariate sequence of degree $2n$. We study the existence of representing measures for $β$ supported in the curve $y=x^{d}$ ($d\ge 1$) in the case when all column dependence relations in the moment matrix $M_n(β)$ are generated by the relation $Y=X^{d}$. We prove that the core variety of $β$, $\mathcal{CV}(L_β)$, is nonempty (equivalently, representing measures exist) if and only if $C$, the partially defined core matrix of $β$, admits a positive, recursively generated completion $C[A]$. Moreover, $\mathcal{CV}(L_β)$ is the entire curve $y=x^{d}$ if and only if there is a positive definite completion $C[A]$. In the remaining case, if there is a measure, it is unique and finitely atomic. For $d = 3$, we use these results to compute the core variety of $β$ and give new characterizations of the existence of representing measures, which complement a result of the first-named author.

math.FA↗

arXiv2025

Matricial Gaussian quadrature rules: singular case

Let $L$ be a linear operator on univariate polynomials of bounded degree taking values in real symmetric matrices, whose moment matrix is positive semidefinite. Assume that $L$ admits a positive matrix-valued representing measure $μ$. Any finitely atomic representing measure with the smallest sum of the ranks of the matricial masses is called minimal. In this paper, we characterize the existence of a minimal representing measure that contains a prescribed atom with a prescribed rank of the corresponding mass, thereby generalizing our recent result, which addresses the same problem in the case where the moment matrix is positive definite. As a corollary, we obtain a constructive, linear-algebraic proof of the strong truncated Hamburger matrix moment problem.

math.FA↗

arXiv2025

Constructive approach to the truncated moment problem on reducible cubic curves: Hyperbolic type relations

In this paper, we solve constructively the bivariate truncated moment problem (TMP) of even degree on reducible cubic curves, where the conic part is a hyperbola. According to the classification from our previous work, these represent three out of nine possible canonical forms of reducible cubic curves after applying an affine linear transformation. The TMP on the union of three parallel lines, the circular and the parabolic type TMP were solved constructively in our previous work, while in this paper we consider three cases of hyperbolic type, i.e., a type without real self-intersection points, a type with a simple real self-intersection point and a type with a double real self-intersection point. In all cases, we also establish bounds on the number of atoms in a minimal representing measure.

math.FA↗

arXiv2025

Cross-positive linear maps, positive polynomials and sums of squares

A linear map $Φ$ between matrix spaces is called cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $\langle U,V\rangle:=\text{Tr}(UV)=0$ implies $\langle Φ(U),V\rangle\geq0$, and is completely cross-positive if all its ampliations $I_n\otimes Φ$ are cross-positive. (Completely) cross-positive maps arise in the theory of operator semigroups, where they are sometimes called exponentially-positive maps, and are also important in the theory of affine processes on symmetric cones in mathematical finance. To each $Φ$ as above a bihomogeneous form is associated by $p_Φ(x,y)=y^TΦ(xx^T)y$. Then $Φ$ is cross-positive if and only if $p_Φ$ is nonnegative on the variety of pairs of orthogonal vectors $\{(x,y)\mid x^Ty=0\}$. Moreover, $Φ$ is shown to be completely cross-positive if and only if $p_Φ$ is a sum of squares modulo the principal ideal $(x^Ty)$. These observations bring the study of cross-positive maps into the powerful setting of real algebraic geometry. Here this interplay is exploited to prove quantitative bounds on the fraction of cross-positive maps that are completely cross-positive. Detailed results about cross-positive maps $Φ$ mapping between $3\times 3$ matrices are given. Finally, an algorithm to produce cross-positive maps that are not completely cross-positive is presented.

math.FA↗

arXiv2025

Matricial Gaussian quadrature rules: nonsingular case

Let $L$ be a linear operator on univariate polynomials of bounded degree, mapping into real symmetric matrices, such that its moment matrix is positive definite. It is known that $L$ admits a finitely atomic positive matrix-valued representing measure $μ$. Any $μ$ with the smallest sum of the ranks of the matricial masses is called minimal. In this paper, we characterize the existence of a minimal representing measure containing a prescribed atom with prescribed rank of the corresponding mass, thus extending a recent result (2020) for the scalar-valued case. As a corollary, we obtain a constructive, linear algebraic proof of the strong truncated Hamburger matrix moment problem in the nonsingular case. The results will be important in the study of the truncated univariate rational matrix moment problem.

math.FA↗

arXiv2025

Extreme values of the mass distribution associated with $d$-quasi-copulas via linear programming

The recent survey published in Fuzzy Sets and Systems nicknamed ``Hitchhiker's Guide'' has raised the rating of quasi-copula problems in the dependence modeling community in spite of the lack of statistical interpretation of quasi-copulas. Some of the open problems listed there were solved, and some conjectured one way or the other. This paper concentrates on the Open Problem 5 of this list concerning bounds on the volume of a $d$--variate quasi-copula. We disprove a recent conjecture published in the same journal on the lower bound of this volume. We also give evidence that the problem is much more difficult than suspected and provide hints about its final solution.

math.ST↗

arXiv2025

Construction of exceptional copositive matrices

An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant $\mathbb{R}^{n}_{\geq 0}$. The cone of copositive matrices contains the cone of matrices which are the sum of a positive semidefinite matrix and a nonnegative one and the latter contains the cone of completely positive matrices. These are the matrices of the form $BB^T$ for some $n\times r$ matrix $B$ with nonnegative entries. The above inclusions are strict for $n\geq5.$ The first main result of this article is a free probability inspired construction of exceptional copositive matrices of all sizes $\geq 5$, i.e., copositive matrices that are not the sum of a positive semidefinite matrix and a nonnegative one. The second contribution of this paper addresses the asymptotic ratio of the volume radii of compact sections of the cones of copositive and completely positive matrices. In a previous work by the authors, it was shown that, by identifying symmetric matrices naturally with quartic even forms, and equipping them with the $L^2$ inner product and the Lebesgue measure, the ratio of the volume radii of sections with a suitably chosen hyperplane is bounded below by a constant independent of $n$ as $n$ tends to infinity. In this paper, we extend this result by establishing an analogous bound when the sections of the cones are unit balls in the Frobenius inner product.

math.FA↗

arXiv2024

Gaussian Quadratures with prescribed nodes via moment theory

Let $μ$ be a positive Borel measure on the real line and let $L$ be the linear functional on univariate polynomials of bounded degree, defined as integration with respect to $μ$. In 2020, Blekherman et al., the characterization of all minimal quadrature rules of $μ$ in terms of the roots of a bivariate polynomial is given and two determinantal representations of this polynomial are established. In particular, the authors solved the question of the existence of a minimal quadrature rule with one prescribed node, leaving open the extension to more prescribed nodes. In this paper, we solve this problem using moment theory as the main tool.

math.FA↗

arXiv2024

The truncated univariate rational moment problem

Given a closed subset $K$ in $\mathbb{R}$, the rational $K$-truncated moment problem ($K$-RTMP) asks to characterize the existence of a positive Borel measure $μ$, supported on $K$, such that a linear functional $\mathcal{L}$, defined on all rational functions of the form $\frac{f}{q}$, where $q$ is a fixed polynomial with all real zeros of even order and $f$ is any real polynomial of degree at most $2k$, is an integration with respect to $μ$. The case of a compact set $K$ was solved by Chandler in 1994, but there is no argument that ensures that $μ$ vanishes on all real zeros of $q$. An obvious necessary condition for the solvability of the $K$-RTMP is that $\mathcal{L}$ is nonnegative on every $f$ satisfying $f|_{K}\geq 0$. If $\mathcal{L}$ is strictly positive on every $0\neq f|_{K}\geq 0$, we add the missing argument from Chandler's solution and also bound the number of atoms in a minimal representing measure. We show by an example that nonnegativity of $\mathcal{L}$ is not sufficient and add the missing conditions to the solution. We also solve the $K$-RTMP for unbounded $K$ and derive the solutions to the strong truncated Hamburger moment problem and the truncated moment problem on the unit circle as special cases.

math.FA↗

arXiv2024

The truncated moment problem on reducible cubic curves I: Parabolic and Circular type relations

In this article we study the bivariate truncated moment problem (TMP) of degree $2k$ on reducible cubic curves. First we show that every such TMP is equivalent after applying an affine linear transformation to one of 8 canonical forms of the curve. The case of the union of three parallel lines was solved in 2022 by the second author, while the degree 6 cases in 2017 by the first author. Second we characterize in terms of concrete numerical conditions the existence of the solution to the TMP on two of the remaining cases concretely, i.e., a union of a line and a circle $y(ay+x^2+y^2)=0, a\in \mathbb{R}\setminus \{0\}$, and a union of a line and a parabola $y(x-y^2)=0$. In both cases we also determine the number of atoms in a minimal representing measure.

math.FA↗