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Harm Derksen

Publications and source records attributed to Harm Derksen.

At least 19 recordsLinked to original sources
arXiv2026

Syzygies of the transfer ideal of the symmetric group

We consider the modular action of the symmetric group $S_n$ on $R = k[x_1,\ldots,x_n]$ when $\mathrm{char}(k) = p \leq n$. We show that the image of the transfer map $R\to R^{S_n}$ is an elimination ideal $J\cap R^{S_n}$, where $J\subset R^{S_n}[t]$ is generated by $p$ polynomials with generic coefficients. The structure of this elimination ideal depends only on the quotient $q$ when writing $n = qp + r$ with unique remainder $0 \leq r < p$, implying that the image of the transfer also enjoys this stability. We conjecture a determinantal presentation of the elimination ideal and prove it in the case that $q = 2$. Furthermore, we exhibit a GL-equivariant, linear minimal free resolution of a certain initial ideal, allowing us to extract the graded Betti numbers of the elimination ideal.

math.AC↗
arXiv2026

Algebraicity of the Brascamp-Lieb constants

We show that the Brascamp-Lieb (BL) constant BL(-,p) is a semi-algebraic function on the set of feasible data. Consequently, it is algebraic in the sense that it satisfies a polynomial relation of the form P(V, BL(V,p))=0 for a non-zero polynomial P. In fact, we establish an analogous statement in the more general setting of quiver BL constants associated to representations of bipartite quivers.

math.RT↗
arXiv2025

Tropical Invariants for Permutation Group Actions

We consider the action of a permutation group $G$ of order $k$ on the tropical polynomial semiring in $n$ variables. We prove that the sub-semiring of invariant polynomials is finitely generated if and only if $G$ is generated by $2$-cycles. There do exist finitely many separating invariants of degree at most $\max\{n,{n\choose 2}\}$. Separating tropical invariants can be used to construct bi-Lipschitz embeddings of the orbit space ${\mathbb R}^n/G$ into Euclidean space. We also show that the invariant polynomials of degree $\leq n p_1p_2\cdots p_k$ generate the semifield of invariant rational tropical functions, where $p_1,p_2,\dots,p_k$ are the first $k$ prime numbers. Most results are also true over arbitrary semirings that are additively idempotent and multiplicatively cancellative.

math.AC↗
arXiv2025

Invariants of finite groups acting on (free) skew fields

Let $M$ be a finitely generated skew field over a ground field $k$, and let $G$ be a finite group of $k$-linear automorphisms of $M$. This paper investigates finite generation of the skew subfield $M^G$ of $G$-invariants in $M$, and relations between the generators. The first main result shows that $M^G$ is finitely generated. Stronger conclusions hold when $M$ is a free skew field, i.e., the universal skew field of fractions of a free algebra. The second main result is the solution of the free Noether problem for non-modular linear group actions: if $G$ acts linearly on the free skew field $M$ on $m$ generators and the characteristic of $k$ does not divide $|G|$, then $M^G$ is the free skew field on $|G|(m-1)+1$ generators. In contrast, a nonlinear action of $Z_2$ on the free skew field $M$ on two generators is presented such that $M^{Z_2}$ is not a free skew field, resolving the free L\"uroth problem. This action also exposes a non-scalar element of $M$ whose centralizer is not a rational field, refuting a conjecture of P. M. Cohn from 1978.

math.RA↗
arXiv2025

Constructive counterexamples to the additivity of minimum output R\'enyi entropy of quantum channels for all $p>1$

We present explicit quantum channels with strictly sub-additive minimum output R\'enyi entropy for all $p>1$, improving upon prior constructions which handled $p>2$. Our example is provided by explicit constructions of linear subspaces with high geometric measure of entanglement. This construction applies in both the bipartite and multipartite settings. As further applications, we use our construction to find entanglement witnesses with many highly negative eigenvalues, and to construct entangled mixed states that remain entangled after perturbation.

quant-ph↗
arXiv2025

Generic orbits, normal bases, and generation degree for fields of rational invariants

For a faithful linear representation $V$ of a finite group $G$ in coprime characteristic, we show that if the field Noether number $\beta_{\mathrm{field}}$ is the minimum $d$ such that the invariant polynomials of degree $\leq d$ generate the field $k(V)^G$ of rational invariants as a field, and the spanning degree $D_\mathrm{span}$ is the minimum $d$ such that the polynomials of degree $\leq d$ span the rational function field $k(V)$ as a vector space over $k(V)^G$, then $\beta_{\mathrm{field}} \leq 2D_\mathrm{span} + 1$, and this is sharp. This generalizes a recent result of Edidin and Katz. We also study $D_\mathrm{span}$. We show that it is related to various quantities previously studied in invariant and representation theory. Dropping the coprime characteristic hypothesis, we prove several basic inequalities, including that it is monotonically nondecreasing in $G$, nonincreasing in $V$, and satisfies $D_\mathrm{span} \leq |G|-1$. The latter refines a recent result of Kollar and Tiep.

math.AC↗
arXiv2025

Estimating the Euclidean distortion of an orbit space

Given a finite-dimensional inner product space $V$ and a group $G$ of isometries, we consider the problem of embedding the orbit space $V/G$ into a Hilbert space in a way that preserves the quotient metric as well as possible. This inquiry is motivated by applications to invariant machine learning. We introduce several new theoretical tools before using them to tackle various fundamental instances of this problem.

math.MG↗
arXiv2024

X-arability of mixed quantum states

The problem of determining when entanglement is present in a quantum system is one of the most active areas of research in quantum physics. Depending on the setting at hand, different notions of entanglement (or lack thereof) become relevant. Examples include separability (of bosons, fermions, and distinguishable particles), Schmidt number, biseparability, entanglement depth, and bond dimension. In this work, we propose and study a unified notion of separability, which we call X-arability, that captures a wide range of applications including these. For a subset (more specifically, an algebraic variety) of pure states X, we say that a mixed quantum state is X-arable if it lies in the convex hull of X. We develop unified tools and provable guarantees for X-arability, which already give new results for the standard separability problem. Our results include: -- An X-tensions hierarchy of semidefinite programs for X-arability (generalizing the symmetric extensions hierarchy for separability), and a new de Finetti theorem for fermionic separability. -- A hierarchy of eigencomputations for optimizing a Hermitian operator over X, with applications to X-tanglement witnesses and polynomial optimization. -- A hierarchy of linear systems for the X-tangled subspace problem, with improved polynomial time guarantees even for the standard entangled subspace problem, in both the generic and worst case settings.

quant-ph↗
arXiv2024

Boosting uniformity in quasirandom groups: fast and simple

We study the communication complexity of multiplying $k\times t$ elements from the group $H=\text{SL}(2,q)$ in the number-on-forehead model with $k$ parties. We prove a lower bound of $(t\log H)/c^{k}$. This is an exponential improvement over previous work, and matches the state-of-the-art in the area. Relatedly, we show that the convolution of $k^{c}$ independent copies of a 3-uniform distribution over $H^{m}$ is close to a $k$-uniform distribution. This is again an exponential improvement over previous work which needed $c^{k}$ copies. The proofs are remarkably simple; the results extend to other quasirandom groups. We also show that for any group $H$, any distribution over $H^{m}$ whose weight-$k$ Fourier coefficients are small is close to a $k$-uniform distribution. This generalizes previous work in the abelian setting, and the proof is simpler.

cs.CC↗
arXiv2024

Pseudorandomness, symmetry, smoothing: II

We prove several new results on the Hamming weight of bounded uniform and small-bias distributions. We exhibit bounded-uniform distributions whose weight is anti-concentrated, matching existing concentration inequalities. This construction relies on a recent result in approximation theory due to Erdéyi (Acta Arithmetica 2016). In particular, we match the classical tail bounds, generalizing a result by Bun and Steinke (RANDOM 2015). Also, we improve on a construction by Benjamini, Gurel-Gurevich, and Peled (2012). We give a generic transformation that converts any bounded uniform distribution to a small-bias distribution that almost preserves its weight distribution. Applying this transformation in conjunction with the above results and others, we construct small-bias distributions with various weight restrictions. In particular, we match the concentration that follows from that of bounded uniformity and the generic closeness of small-bias and bounded-uniform distributions, answering a question by Bun and Steinke (RANDOM 2015). Moreover, these distributions are supported on only a constant number of Hamming weights. We further extend the anti-concentration constructions to small-bias distributions perturbed with noise, a class that has received much attention recently in derandomization. Our results imply (but are not implied by) a recent result of the authors (CCC 2024), and are based on different techniques. In particular, we prove that the standard Gaussian distribution is far from any mixture of Gaussians with bounded variance.

cs.CC↗
arXiv2024

Pseudorandomness, symmetry, smoothing: I

We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that 1) achieve an optimal lower bound on their statistical distance to any bounded-uniform distribution. This closes a line of research initiated by Alon, Goldreich, and Mansour in 2003, and improves on a result by O'Donnell and Zhao. 2) have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke. 3) rule out a popular paradigm for constructing pseudorandom generators, originating in a 1989 work by Ajtai and Wigderson. This again answers a question raised by several researchers. For branching programs, our result matches a bound by Forbes and Kelley. Our small-bias distributions above are symmetric. We show that the xor of any two symmetric small-bias distributions fools any bounded function. Hence our examples cannot be extended to the xor of two small-bias distributions, another popular paradigm whose power remains unknown. We also generalize and simplify the proof of a result of Bazzi.

cs.CC↗
arXiv2023

Quasirandom groups enjoy interleaved mixing

Let $G$ be a group such that any non-trivial representation has dimension at least $d$. Let $X=(X_{1},X_{2},\ldots,X_{t})$ and $Y=(Y_{1},Y_{2},\ldots,Y_{t})$ be distributions over $G^{t}$. Suppose that $X$ is independent from $Y$. We show that for any $g\in G$ we have $|\mathbb{P}[X_{1}Y_{1}X_{2}Y_{2}\cdots X_{t}Y_{t}=g]-1/|G||\le\frac{|G|^{2t-1}}{d^{t-1}}\sqrt{\mathbb{E}_{h\in G^{t}}X(h)^{2}}\sqrt{\mathbb{E}_{h\in G^{t}}Y(h)^{2}}.$ Our results generalize, improve, and simplify previous works.

math.CO↗
arXiv2023

The capacity of quiver representations and the Anantharam-Jog-Nair inequality

The Anantharam-Jog-Nair inequality [AJN22] in Information Theory provides a unifying approach to the information-theoretic form of the Brascamp-Lieb inequality [CCE09] and the Entropy Power inequality [ZF93]. In this paper, we use methods from Quiver Invariant Theory [CD21] to study Anantharam-Jog-Nair inequalities with integral exponents. For such an inequality, we first view its input datum as a quiver datum and show that the best constant that occurs in the Anantharam-Jog-Nair inequality is of the form $-{1\over 2}\log (\mathbf{cap}(V,σ))$ where $\mathbf{cap}(V, σ)$ is the capacity of a quiver datum $(V, σ)$ of a complete bipartite quiver. The general tools developed in [CD21], when applied to complete bipartite quivers, yield necessary and sufficient conditions for: (1) the finiteness of the Anantharam-Jog-Nair best constants; and (2) the existence of Gaussian extremizers. These results recover some of the main results in [AJN22] and [ACZ22]. In addition, we characterize gaussian-extremizable data in terms of semi-simple data, and provide a general character formula for the Anatharam-Jog-Nair constants. Furthermore, our quiver invariant theoretic methods lead to necessary and sufficient conditions for the uniqueness of Gaussian extremizers. This answers the third and last question left unanswered in [AJN22].

cs.IT↗
arXiv2023

Tensor Denoising via Amplification and Stable Rank Methods

Tensors in the form of multilinear arrays are ubiquitous in data science applications. Captured real-world data, including video, hyperspectral images, and discretized physical systems, naturally occur as tensors and often come with attendant noise. Under the additive noise model and with the assumption that the underlying clean tensor has low rank, many denoising methods have been created that utilize tensor decomposition to effect denoising through low rank tensor approximation. However, all such decomposition methods require estimating the tensor rank, or related measures such as the tensor spectral and nuclear norms, all of which are NP-hard problems. In this work we leverage our previously developed framework of $\textit{tensor amplification}$, which provides good approximations of the spectral and nuclear tensor norms, to denoising synthetic tensors of various sizes, ranks, and noise levels, along with real-world tensors derived from physiological signals. We also introduce two new notions of tensor rank -- $\textit{stable slice rank}$ and $\textit{stable }$$X$$\textit{-rank}$ -- and new denoising methods based on their estimation. The experimental results show that in the low rank context, tensor-based amplification provides comparable denoising performance in high signal-to-noise ratio (SNR) settings and superior performance in noisy (i.e., low SNR) settings, while the stable $X$-rank method achieves superior denoising performance on the physiological signal data.

cs.LG↗
arXiv2023

Ranks of linear matrix pencils separate simultaneous similarity orbits

This paper solves the two-sided version and provides a counterexample to the general version of the 2003 conjecture by Hadwin and Larson. Consider evaluations of linear matrix pencils $L=T_0+x_1T_1+\cdots+x_mT_m$ on matrix tuples as $L(X_1,\dots,X_m)=I\otimes T_0+X_1\otimes T_1+\cdots+X_m\otimes T_m$. It is shown that ranks of linear matrix pencils constitute a collection of separating invariants for simultaneous similarity of matrix tuples. That is, $m$-tuples $A$ and $B$ of $n\times n$ matrices are simultaneously similar if and only if the ranks of $L(A)$ and $L(B)$ are equal for all linear matrix pencils $L$ of size $mn$. Variants of this property are also established for symplectic, orthogonal, unitary similarity, and for the left-right action of general linear groups. Furthermore, a polynomial time algorithm for orbit equivalence of matrix tuples under the left-right action of special linear groups is deduced.

math.RA↗
arXiv2022

Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors

Since the seminal works of Strassen and Valiant it has been a central theme in algebraic complexity theory to understand the relative complexity of algebraic problems, that is, to understand which algebraic problems (be it bilinear maps like matrix multiplication in Strassen's work, or the determinant and permanent polynomials in Valiant's) can be reduced to each other (under the appropriate notion of reduction). In this paper we determine precisely how many independent scalar multiplications can be reduced to a given bilinear map (this number is called the subrank, and extends the concept of matrix diagonalization to tensors), for essentially all (i.e. generic) bilinear maps. Namely, we prove for a generic bilinear map $T : V \times V \to V$ where $\dim(V) = n$ that $θ(\sqrt{n})$ independent scalar multiplications can be reduced to $T$. Our result significantly improves on the previous upper bound from the work of Strassen (1991) and Bürgisser (1990) which was $n^{2/3 + o(1)}$. Our full result is much more general and applies not only to bilinear maps and 3-tensors but also to $k$-tensors, for which we find that the generic subrank is $θ(n^{1/(k-1)})$. Moreover, as an application we prove that the subrank is not additive under the direct sum. The subrank plays a central role in several areas of complexity theory (matrix multiplication algorithms, barrier results) and combinatorics (e.g., the cap set problem and sunflower problem). As a consequence of our result we obtain several large separations between the subrank and tensor methods that have received much interest recently, notably the slice rank (Tao, 2016), analytic rank (Gowers--Wolf, 2011; Lovett, 2018; Bhrushundi--Harsha--Hatami--Kopparty--Kumar, 2020), geometric rank (Kopparty--Moshkovitz--Zuiddam, 2020), and G-stable rank (Derksen, 2020).

cs.CC↗