Symmetric subrank and its border analogue

Abstract: The symmetric subrank of homogeneous polynomial is the largest number of terms in a diagonal form to which it can be specialized by a (typically non-invertible) linear variable substitution. Building on earlier work by Derksen-Makam-Zuiddam and Biaggi-Chang-Draisma-Rupniewski for ordinary tensors, we determine the asymptotic behavior of symmetric subrank and symmetric border subrank of degree-d forms as the number of variables tends to infinity. Furthermore, by using results from geometric invariant theory we show that for cubic (resp. quartic) forms the symmetric subrank and symmetric border subrank coincide if the latter is at most three (resp. two).

• The paper introduces symmetric subrank and its border analogue, establishing new complexity bounds for homogeneous polynomials.
• It employs geometric invariant theory and syzygetic arguments to derive precise asymptotic growth rates in increasing dimensions.
• It contrasts symmetric and non-symmetric tensor cases by providing explicit examples where border subrank exceeds strict subrank.

Symmetric Subrank and Its Border Analogue: Foundations and Asymptotics

Introduction and Definitions

The paper "Symmetric subrank and its border analogue" (2604.12801) extends the algebraic complexity analysis of tensorial and polynomial structures by formalizing and investigating the notion of symmetric subrank and its border analogue for homogeneous polynomials (equivalently, symmetric tensors). Building on the non-symmetric tensor subrank results of Derksen-Makam-Zuiddam and Biaggi-Chang-Draisma-Rupniewski, the authors develop the symmetric analogues and characterize their behavior, especially in the asymptotic regime as the ambient dimension grows.

Let $V$ be an $n$-dimensional vector space over an algebraically closed field $K$, and $S^d V$ denote the $d$-th symmetric power, i.e., the space of degree-$d$ homogeneous polynomials (or symmetric tensors) on $V$.

The symmetric subrank $Q_s(f)$ of $f \in S^d V$ is the maximal $r$ such that, after a (possibly non-invertible) linear transformation, $n$0 can be specialized to the "unit tensor" $n$1 (the diagonal sum of $n$2-th powers), that is,

$n$3

Its border analogue, the symmetric border subrank $n$4, is the maximal $n$5 such that $n$6 lies in the Zariski closure of the images of $n$7 under $n$8-linear maps $n$9:

$K$0

These are symmetric analogues of Strassen's subrank and border subrank for general tensors, which are important complexity invariants in understanding bilinear map composition, tensorial evaluation, and, by extension, the hardness of algorithms such as matrix multiplication.

Asymptotic Behavior in Increasing Ambient Dimension

The main asymptotic result proven is that, for fixed degree $K$1 and dimension $K$2, the generic symmetric subrank $K$3 of a random tensor in $K$4 grows as

$K$5

This matches the form of previously obtained non-symmetric results for tensor subrank, but with explicit dependence on $K$6, reflecting the structural constraints of symmetry.

Generic Symmetric Border Subrank

Analogously, the generic symmetric border subrank exhibits identical asymptotic growth,

$K$7

The constants differ, but the exponent is sharp. The proof uses an adaptation of geometric invariant theory techniques and syzygetic arguments (specifically, leveraging Hochster–Laksov results on syzygies and the fiber dimension theorem for estimating parameter counts).

Comparison and Algebraic Invariant Theory

Hilbert-Mumford Subrank Tightness

A significant structural result is that, for symmetric tensors, the symmetric subrank and the symmetric Hilbert-Mumford subrank always coincide:

$K$8

where $K$9 is defined via optimal degeneration to $S^d V$0-orbits of $S^d V$1 through one-parameter subgroup actions in the sense of geometric invariant theory. This is in contrast to the non-symmetric case, where Hilbert-Mumford subrank can exceed subrank. Therefore, border techniques premised on additional degenerations do not differentiate between these ranks for symmetric tensors.

Strict Inequality: Border vs. Symmetric Subrank

Despite the tightness above, the authors construct explicit quintic forms (for $S^d V$2, $S^d V$3) where the symmetric border subrank strictly exceeds the symmetric subrank:

• For $S^d V$4,
  • $S^d V$5 but $S^d V$6.

Thus, while the generic values of subrank and border subrank align asymptotically, there exist forms for which $S^d V$7, but minimal such counterexamples require higher degree and dimension.

Low-Rank Coincidence for Cubics and Quartics

The study includes a detailed geometric analysis for low values of the border subrank, particularly for cubic and quartic forms:

• For cubics ($S^d V$8) with $S^d V$9,
• and quartics ($d$0) with $d$1, it is shown that

$d$2

This result leverages geometric invariant theory and properties of linear sections of hypersurfaces, and is tied to the structure of the moduli spaces of degree $d$3 forms in projective dimension $d$4 when these moduli have minimal complexity (i.e., isomorphic to affine space).

Geometric and Algebraic Tools

The theory throughout the paper uses a blend of:

• Geometric invariant theory: Analyzing $d$5-orbits and orbit closures in $d$6.
• Algebraic geometry: Properties of projective hypersurfaces and syzygies.
• Combinatorics: Dimension counting via Grassmannians and parameter spaces for linear maps.
• Representation theory and symmetrization: Identifying the structure of $d$7 and orbit dimensions, especially with reference to field characteristic constraints.

Practical and Theoretical Implications

The characterization of symmetric subrank and its border analogues sharpens the understanding of symmetrically structured polynomial complexity, which is fundamental for:

• Algebraic complexity theory, particularly lower bounds for symmetric functions and Waring-type decompositions.
• Computational algebra, such as algorithms for symmetric tensor decomposition.
• Theoretical computer science, in context of optimizing computation over commutative algebraic structures.

It also demonstrates that, for symmetric tensors, border degenerations do not generally yield higher generic subrank compared to strict subrank, although exceptions occur at higher degrees. Thus, existing lower bound techniques that exploit border rank separation fail for symmetric subrank generically.

Future developments may focus on:

• Precise determination of the multiplicative constants for border subrank, narrowing the small asymptotic constant gap proved.
• Extending the geometric approach to higher degrees and finding explicit families realizing the gap between subrank and border subrank.
• Connections to algebraic statistics and entanglement theory, where symmetric tensors play a role in describing highly structured probability distributions or multipartite quantum states.

Conclusion

This work rigorously establishes the asymptotic scaling and tightness properties of symmetric subrank and its border analogue. For cubic and quartic forms, low border subranks coincide with strict subrank, while for higher degree polynomials, explicit counterexamples exist. The results clarify the extent to which border techniques can enhance classical algebraic complexity measures in the symmetric setting and lay the groundwork for further explorations in symmetric tensor complexity and invariant theory.