arXiv subjects

Ian Orzel

Publications and source records attributed to Ian Orzel.

4 recordsLinked to original sources

arXiv2026

Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?

The complexity of bilinear maps (equivalently, of $3$-mode tensors) has been studied extensively, most notably in the context of matrix multiplication. While circuit complexity and tensor rank coincide asymptotically for $3$-mode tensors, this correspondence breaks down for $d \geq 4$ modes. As a result, the complexity of $d$-mode tensors for larger fixed $d$ remains poorly understood, despite its relevance, e.g., in fine-grained complexity. Our paper explores this intermediate regime. First, we give a "graph-theoretic" proof of Strassen's $2\omega/3$ bound on the asymptotic rank exponent of $3$-mode tensors. Our proof directly generalizes to an upper bound of $(d-1)\omega/3$ for $d$-mode tensors. Using refined techniques available only for $d\geq 4$ modes, we improve this bound beyond the current state of the art for $\omega$. We also obtain a bound of $d/2+1$ on the asymptotic exponent of circuit complexity of generic $d$-mode tensors and optimized bounds for $d \in \{4,5\}$. To the best of our knowledge, asymptotic circuit complexity (rather than rank) of tensors has not been studied before. To obtain a robust theory, we first ask whether low complexity of $T$ and $U$ imply low complexity of their Kronecker product $T \otimes U$. While this crucially holds for rank (and thus for circuit complexity in $3$ modes), we show that assumptions from fine-grained complexity rule out such a submultiplicativity for the circuit complexity of tensors with many modes. In particular, assuming the Hyperclique Conjecture, this failure occurs already for $d=8$ modes. Nevertheless, we can salvage a restricted notion of submultiplicativity. From a technical perspective, our proofs heavily make use of the graph tensors $T_H$, as employed by Christandl and Zuiddam ({\em Comput.~Complexity}~28~(2019)~27--56) and [...]

cs.CC↗

arXiv2025

Computing the Elementary Symmetric Polynomials in Positive Characteristics

We first extend the results of Chatterjee,Kumar,Shi,Volk(Computational Complexity 2022) by showing that the degree $d$ elementary symmetric polynomials in $n$ variables have formula lower bounds of $\Omega(d(n-d))$ over fields of positive characteristic. Then, we show that the results of the universality of linear projections of elementary symmetric polynomials from Shpilka(JCSS 2002) and of border fan-in two $\Sigma\Pi\Sigma$ circuits from Kumar(ACM TOCT 2020) over zero characteristic fields do not extend to fields of positive characteristic. In particular, we show that *There are polynomials that cannot be represented as linear projections of the elementary symmetric polynomials(in fact, we show linear lower bounds over the size of the sum of such linear projections) and *There are polynomials that cannot be computed by border depth-$3$ circuits of top fan-in $k$, called $\overline{\Sigma^{[k]}\Pi\Sigma}$, for $k = o(n)$. To prove the first result, we consider a geometric property of the elementary symmetric polynomials, namely, the set of all points in which the polynomial and all of its first-order partial derivatives vanish. It was previously shown that the dimension of this space was exactly $d-2$ for fields of zero characteristic. We extend this to fields of positive characteristic by showing that this dimension must be between $d-2$ and $d-1$. In fact, we provide some criterion where it is $d-2$ and others where it is $d-1$. Then, to consider the border top fan-in of linear projections of the elementary symmetric polynomials and border depth-$3$ circuits(sometimes called border affine Chow rank), we show that it is sufficient to consider the border top fan-in of the sum of such linear projections of the elementary symmetric polynomials. This is done by an explicit construction of a 'metapolynomial,' meaning that this result also applies in the border setting.

cs.CC↗

arXiv2025

Bessmertny\u{i} realizations of symmetric multivariate rational matrix functions over any field

In this paper, we prove the following. First, every square matrix whose entries are multivariable rational functions over a field $\mathbb{F}$ has a Bessmertny\u{i} realization, i.e., is the Schur complement of an affine linear square matrix pencil with coefficients in $\mathbb{F}$. Second, if the matrix is also symmetric and the characteristic of the field $\mathbb{F}$ is not two then it has a symmetric Bessmertny\u{i} realization (i.e., the pencil can be chosen to consist of symmetric matrices) and counterexamples are given to prove this statement is false in general for fields of characteristic two. Third, for fields of characteristic two (e.g., binary or Boolean field), we completely characterize those functions that have a symmetric Bessmertny\u{i} realization. Finally, analogous results hold when restricted to the class of homogeneous degree-one rational functions. To solve these realization problems, i.e., finding such structured Bessmertny\u{i} realizations for a given multivariate rational function, we use state-space methods from systems theory to produce realizations for algebraic operations on Schur complements such as sums, products, inverses, and symmetrization, which become the elementary building blocks of our constructions. Further complications arise over fields of characteristic two, so a large part of the paper is devoted to developing additional methods to decide if the symmetric realization problem can be solved and, if so, to construction the symmetric realization for a given symmetric rational matrix function. Our motivations are discussed in the context of multidimensional linear systems theory on generalizing state-space representations for rational functions including the Givone-Roesser and Fornasini-Marchesini realizations.

math.RA↗

arXiv2025

The Algebraic Cost of a Boolean Sum

It is a well-known fact that the permanent polynomial is complete for the complexity class VNP, and it is largely suspected that the determinant does not share this property, despite its similar expression. We study the question of why the VNP-completeness proof of the permanent fails for the determinant. We isolate three fundamental properties that are sufficient to prove a polynomial sequence is VNP-hard, of which two are shared by both the permanent and the determinant. We proceed to show that the permanent satisfies the third property, which we refer to as the ``cost of a boolean sum," while the determinant does not, showcasing the fundamental difference between the polynomial families. We further note that this differentiation also applies in the border complexity setting and that our results apply for counting complexity.

cs.CC↗