arXiv subjects

Sizhuo Yan

Publications and source records attributed to Sizhuo Yan.

5 recordsLinked to original sources

arXiv2026

Noncommutative Rational Sums of Squares in Free Algebras

This paper introduces rational sums of squares for symmetric noncommutative polynomials in free algebras. Under the assumption that the polynomial admits a strictly positive scalar self-adjoint evaluation, our main result gives an operator-theoretic characterization of this class: a polynomial is a rational sum of squares if and only if there exists a degree bound such that, for every self-adjoint operator evaluation satisfying a natural nondegeneracy condition, the compression of the evaluated polynomial to the associated finite-dimensional cyclic subspace is not negative definite. We also prove that sums of Hermitian squares form a proper subset of rational sums of squares when the free algebra has at least two generators, whereas the two classes coincide for homogeneous polynomials. Finally, we show that the set of rational sums of squares is nonconvex when there are at least three generators, and that its complement is also nonconvex.

math.RA↗

arXiv2026

A Homogeneous Nullstellensatz for Joint Invariant Subspaces

Jurij Vol\v{c}i\v{c} conjectured that a noncommutative polynomial $g$ belongs to the unital $\mathbb{K}$-algebra generated by finitely many noncommutative polynomials if and only if, for matrices of every size, every joint invariant subspace of the evaluations of the generators is also invariant under the evaluation of $g$. In this paper, we establish a homogeneous Nullstellensatz for joint invariant subspaces by proving that this equivalence holds whenever the generators are homogeneous. In contrast, we demonstrate that the statement fails in the general case, thereby settling the conjecture completely.

math.RA↗

arXiv2023

The non-Archimedean Nirgendsnegativsemidefinitheitsstellensatz is not true

Klep and Schweighofer asked whether the Nirgendsnegativsemide-finitheitsstellensatz holds for a symmetric noncommutative polynomial whose evaluations at bounded self-adjoint operators on any nontrivial Hilbert space are not negative semidefinite. We provide an example to show the open problem has a negative answer.

math.OA↗

arXiv2023

A Characterization of Perfect Strategies for Mirror Games

We associate mirror games with the universal game algebra and use the *-representation to describe quantum commuting operator strategies. We provide an algebraic characterization of whether or not a mirror game has perfect commuting operator strategies. This new characterization uses a smaller algebra introduced by Paulsen and others for synchronous games and the noncommutative Nullstellensatz developed by Cimpric, Helton and collaborators. An algorithm based on noncommutative Gröbner basis computation and semidefinite programming is given for certifying that a given mirror game has no perfect commuting operator strategies.

math.OA↗

arXiv2022

Extensions of S-Lemma for Noncommutative Polynomials

We consider the problem of extending the classical S-lemma from commutative case to noncommutative cases. We show that a symmetric quadratic homogeneous matrix-valued polynomial is positive semidefinite if and only if its coefficient matrix is positive semidefinite. Then we extend the S-lemma to three kinds of noncommutative polynomials: noncommutative polynomials whose coefficients are real numbers, matrix-valued noncommutative polynomials and hereditary polynomials.

math.OC↗