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Amir Shpilka

Publications and source records attributed to Amir Shpilka.

At least 19 recordsLinked to original sources
arXiv2026

Tensor Reconstruction Beyond Constant Rank

We give reconstruction algorithms for subclasses of depth-3 arithmetic circuits. In particular, we obtain the first efficient algorithm for finding tensor rank, and an optimal tensor decomposition as a sum of rank-one tensors, when given black-box access to a tensor of super-constant rank. We obtain the following results: 1. A randomized algorithm that reconstructs polynomials computed by multilinear $Σ^{k]}\prod^{[d]}Σ$ circuits in time $\mathsf{poly}(n,d,c) \cdot k^{k^{k^{k^{O(k)}}}}$ 2. A randomized algorithm that reconstructs polynomials computed by set-multilinear $Σ^{k]}\prod^{[d]}Σ$ circuits in time $\mathsf{poly}(n,d,c) \cdot k^{k^{k^{k^{O(k)}}}}$, where $c=\log q$ if $\mathbb{F}=\mathbb{F}_q$ is a finite field, and $c$ equals the maximum bit complexity of any coefficient of $f$ if $\mathbb{F}$ is infinite. Prior to our work, polynomial time algorithms for the case when the rank, $k$, is constant, were given by Bhargava, Saraf and Volkovich [BSV21]. Another contribution of this work is correcting an error from a paper of Karnin and Shpilka [KS09] that affected Theorem 1.6 of [BSV21]. Consequently, the results of [KS09, BSV21] continue to hold, with a slightly worse setting of parameters. For fixing the error we study the relation between syntactic and semantic ranks of $ΣΠΣ$ circuits. We obtain our improvement by introducing a technique for learning rank preserving coordinate-subspaces. [KS09] and [BSV21] tried all choices of finding the "correct" coordinates, which led to having a fast growing function of $k$ at the exponent of $n$. We find these spaces in time that is growing fast with $k$, yet it is only a fixed polynomial in $n$.

cs.CC↗
arXiv2026

On Factorization of Sparse Polynomials of Bounded Individual Degree

We study sparse polynomials with bounded individual degree and their factors, obtaining the following structural and algorithmic results. 1. A deterministic polynomial-time algorithm to find all sparse divisors of a sparse polynomial of bounded individual degree, together with the first upper bound on the number of non-monomial irreducible factors of such polynomials. 2. A $\mathrm{poly}(n,s^{d\log \ell})$-time algorithm that recovers $\ell$ irreducible $s$-sparse polynomials of individual degree at most $d$ from blackbox access to their (not necessarily sparse) product. This partially resolves a question of Dutta-Sinhababu-Thierauf (RANDOM 2024). In particular, if $\ell=O(1)$ the algorithm runs in polynomial time. 3. Deterministic algorithms for factoring a product of $s$-sparse polynomials of individual degree $d$ from blackbox access. Over fields of characteristic zero or sufficiently large characteristic the runtime is $\mathrm{poly}(n,s^{d^3\log n})$; over arbitrary fields it is $\mathrm{poly}(n,(d^2)!,s^{d^5\log n})$. This improves Bhargava-Saraf-Volkovich (JACM 2020), which gives $\mathrm{poly}(n,s^{d^7\log n})$ time for a single sparse polynomial. For a single sparse input we obtain $\mathrm{poly}(n,s^{d^2\log n})$ time. 4. Given blackbox access to a product of factors of sparse polynomials of bounded individual degree, we give a deterministic polynomial-time algorithm to find all irreducible sparse multiquadratic factors with multiplicities. This generalizes the algorithms of Volkovich (RANDOM 2015, 2017) and extends the complete-power test of Bisht-Volkovich (CC 2025). To handle arbitrary fields we introduce a notion of primitive divisors that removes characteristic assumptions from most of our algorithms.

cs.CC↗
arXiv2026

Polynomial Identity Testing and Reconstruction for Depth-4 Powering Circuits of High Degree

We study deterministic polynomial identity testing (PIT) and reconstruction algorithms for depth-$4$ arithmetic circuits of the form \[ Σ^{[r]}\!\wedge^{[d]}\!Σ^{[s]}\!Π^{[δ]}. \] This model generalizes Waring decompositions and diagonal circuits, and captures sums of powers of low-degree sparse polynomials. Specifically, each circuit computes a sum of $r$ terms, where each term is a $d$-th power of an $s$-sparse polynomial of degree $δ$. This model also includes algebraic representations that arise in tensor decomposition and moment-based learning tasks such as mixture models and subspace learning. We give deterministic worst-case algorithms for PIT and reconstruction in this model. Our PIT construction applies when $d>r^2$ and yields explicit hitting sets of size $O(r^4 s^4 n^2 d δ^3)$. The reconstruction algorithm runs in time $\textrm{poly}(n,s,d)$ under the condition $d=Ω(r^4δ)$, and in particular it tolerates polynomially large top fan-in $r$ and bottom degree $δ$. Both results hold over fields of characteristic zero and over fields of sufficiently large characteristic. These algorithms provide the first polynomial-time deterministic solutions for depth-$4$ powering circuits with unbounded top fan-in. In particular, the reconstruction result improves upon previous work which required non-degeneracy or average-case assumptions. The PIT construction relies on the ABC theorem for function fields (Mason-Stothers theorem), which ensures linear independence of high-degree powers of sparse polynomials after a suitable projection. The reconstruction algorithm combines this with Wronskian-based differential operators, structural properties of their kernels, and a robust version of the Klivans-Spielman hitting set.

cs.CC↗
arXiv2025

Discreteness of asymptotic tensor ranks

Tensor parameters that are amortized or regularized over large tensor powers, often called "asymptotic" tensor parameters, play a central role in several areas including algebraic complexity theory (constructing fast matrix multiplication algorithms), quantum information (entanglement cost and distillable entanglement), and additive combinatorics (bounds on cap sets, sunflower-free sets, etc.). Examples are the asymptotic tensor rank, asymptotic slice rank and asymptotic subrank. Recent works (Costa-Dalai, Blatter-Draisma-Rupniewski, Christandl-Gesmundo-Zuiddam) have investigated notions of discreteness (no accumulation points) or "gaps" in the values of such tensor parameters. We prove a general discreteness theorem for asymptotic tensor parameters of order-three tensors and use this to prove that (1) over any finite field (and in fact any finite set of coefficients in any field), the asymptotic subrank and the asymptotic slice rank have no accumulation points, and (2) over the complex numbers, the asymptotic slice rank has no accumulation points. Central to our approach are two new general lower bounds on the asymptotic subrank of tensors, which measures how much a tensor can be diagonalized. The first lower bound says that the asymptotic subrank of any concise three-tensor is at least the cube-root of the smallest dimension. The second lower bound says that any concise three-tensor that is "narrow enough" (has one dimension much smaller than the other two) has maximal asymptotic subrank. Our proofs rely on new lower bounds on the maximum rank in matrix subspaces that are obtained by slicing a three-tensor in the three different directions. We prove that for any concise tensor, the product of any two such maximum ranks must be large, and as a consequence there are always two distinct directions with large max-rank.

cs.CC↗
arXiv2025

On the Complexity of Hazard-Free Formulas

This paper studies the hazard-free formula complexity of Boolean functions. Our first result shows that unate functions are the only Boolean functions for which the monotone formula complexity of the hazard-derivative equals the hazard-free formula complexity of the function itself. Consequently, they are the only functions for which the hazard-derivative approach of Ikenmeyer et al. (J. ACM, 2019) yields optimal bounds. Our second result proves that the hazard-free formula complexity of random Boolean functions is at most $2^{(1+o(1))n}$. Prior to this, no better upper bound than $O(3^n)$ was known. Notably, unlike in the general case of Boolean circuits and formulas, where the typical complexity is derived from that of the multiplexer function with $n$-bit selector, the hazard-free formula complexity of a random function is smaller than the optimal hazard-free formula for the multiplexer by an exponential factor in $n$. We provide two proofs of this fact. The first is direct, bounding the number of prime implicants of a random Boolean function and using this bound to construct a DNF of the claimed size. The second introduces a new and independently interesting result: a weak converse to the hazard-derivative lower bound method, which gives an upper bound on the hazard-free complexity of a function in terms of the monotone complexity of a subset of its hazard-derivatives. Additionally, we explore the hazard-free formula complexity of block composition of Boolean functions and obtain a result in the hazard-free setting that is analogous to a result of Karchmer, Raz, and Wigderson (Computational Complexity, 1995) in the monotone setting. We show that our result implies a stronger lower bound on the hazard-free formula depth of the block composition of the set covering function with the multiplexer function than the bound obtained via the hazard-derivative method.

cs.CC↗
arXiv2025

A New Bound on Cofactors of Sparse Polynomials

We prove that for polynomials $f, g, h \in \mathbb{Z}[x]$ satisfying $f = gh$ and $f(0) \neq 0$, the $\ell_2$-norm of the cofactor $h$ is bounded by $\|h\|_2 \leq \|f\|_1 \cdot\left( \widetilde{O}\left(\|g\|_0^3 \frac{\text{deg }{(f)}^2}{\sqrt{\text{deg }{(g)}}}\right)\right)^{\|g\|_0 - 1}$, where $\|g\|_0$ is the number of nonzero coefficients of $g$ (its sparsity). We also obtain similar results for polynomials over $\mathbb{C}$. This result significantly improves upon previously known exponential bounds (in $\text{deg }{(f)}$) for general polynomials. It further implies that, under exact division, the polynomial division algorithm runs in quasi-linear time with respect to the input size and the number of terms in the quotient $h$. This resolves a long-standing open problem concerning the exact divisibility of sparse polynomials. In particular, our result demonstrates a quadratic separation between the runtime (and representation size) of exact and non-exact divisibility by sparse polynomials. Notably, prior to our work, it was not even known whether the representation size of the quotient polynomial could be bounded by a sub-quadratic function of its number of terms, specifically of $\text{deg }{(f)}$.

cs.SC↗
arXiv2025

Improved Debordering of Waring Rank

We prove that if a degree-$d$ homogeneous polynomial $f$ has border Waring rank $\underline{\mathrm{WR}}({f}) = r$, then its Waring rank is bounded by \[ {\mathrm{WR}}({f}) \leq d \cdot r^{O(\sqrt{r})}. \] This result significantly improves upon the recent bound ${\mathrm{WR}}({f}) \leq d \cdot 4^r$ established in [Dutta, Gesmundo, Ikenmeyer, Jindal, and Lysikov, STACS 2024], which itself was an improvement over the earlier bound ${\mathrm{WR}}({f}) \leq d^r$.

cs.CC↗
arXiv2024

Optimal Two-Dimensional Reed--Solomon Codes Correcting Insertions and Deletions

Constructing Reed--Solomon (RS) codes that can correct insertions and deletions (insdel errors) has been considered in numerous recent works. For the special case of two-dimensional RS-codes, it is known [CST23] that an $[n,2]_q$ RS-code that can correct from $n-3$ insdel errors satisfies that $q=Ω(n^3)$. On the other hand, there are several known constructions of $[n,2]_q$ RS-codes that can correct from $n-3$ insdel errors, where the smallest field size is $q=O(n^4)$. In this short paper, we construct $[n,2]_q$ Reed--Solomon codes that can correct $n-3$ insdel errors with $q=O(n^3)$, thereby resolving the minimum field size needed for such codes.

cs.IT↗
arXiv2024

Lower Bounds for Set-Multilinear Branching Programs

In this paper, we prove super-polynomial lower bounds for the model of \emph{sum of ordered set-multilinear algebraic branching programs}, each with a possibly different ordering ($\sum \mathsf{smABP}$). Specifically, we give an explicit $nd$-variate polynomial of degree $d$ such that any $\sum \mathsf{smABP}$ computing it must have size $n^{ω(1)}$ for $d$ as low as $ω(\log n)$. Notably, this constitutes the first such lower bound in the low degree regime. Moreover, for $d = \poly(n)$, we demonstrate an exponential lower bound. This result generalizes the seminal work of Nisan (STOC, 1991), which proved an exponential lower bound for a single ordered set-multilinear ABP. The significance of our lower bounds is underscored by the recent work of Bhargav, Dwivedi, and Saxena (to appear in TAMC, 2024), which showed that super-polynomial lower bounds against a sum of ordered set-multilinear branching programs -- for a polynomial of sufficiently low degree -- would imply super-polynomial lower bounds against general ABPs, thereby resolving Valiant's longstanding conjecture that the permanent polynomial can not be computed efficiently by ABPs. More precisely, their work shows that if one could obtain such lower bounds when the degree is bounded by $O(\log n/ \log \log n)$, then it would imply super-polynomial lower bounds against general ABPs. Our results strengthen the works of Arvind \& Raja (Chic. J. Theor. Comput. Sci., 2016) and Bhargav, Dwivedi \& Saxena (to appear in TAMC, 2024), as well as the works of Ramya \& Rao (Theor. Comput. Sci., 2020) and Ghoshal \& Rao (International Computer Science Symposium in Russia, 2021), each of which established lower bounds for related or restricted versions of this model. They also strongly answer a question from the former two, which asked to prove super-polynomial lower bounds for general $\sum \mathsf{smABP}$.

cs.CC↗
arXiv2022

On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts

We say that two given polynomials $f, g \in R[X]$, over a ring $R$, are equivalent under shifts if there exists a vector $a \in R^n$ such that $f(X+a) = g(X)$. Grigoriev and Karpinski (FOCS 1990), Lakshman and Saunders (SICOMP, 1995), and Grigoriev and Lakshman (ISSAC 1995) studied the problem of testing polynomial equivalence of a given polynomial to any $t$-sparse polynomial, over the rational numbers, and gave exponential time algorithms. In this paper, we provide hardness results for this problem. Formally, for a ring $R$, let $\mathrm{SparseShift}_R$ be the following decision problem. Given a polynomial $P(X)$, is there a vector $a$ such that $P(X+a)$ contains fewer monomials than $P(X)$. We show that $\mathrm{SparseShift}_R$ is at least as hard as checking if a given system of polynomial equations over $R[x_1,\ldots, x_n]$ has a solution (Hilbert's Nullstellensatz). As a consequence of this reduction, we get the following results. 1. $\mathrm{SparseShift}_\mathbb{Z}$ is undecidable. 2. For any ring $R$ (which is not a field) such that $\mathrm{HN}_R$ is $\mathrm{NP}_R$-complete over the Blum-Shub-Smale model of computation, $\mathrm{SparseShift}_{R}$ is also $\mathrm{NP}_{R}$-complete. In particular, $\mathrm{SparseShift}_{\mathbb{Z}}$ is also $\mathrm{NP}_{\mathbb{Z}}$-complete. We also study the gap version of the $\mathrm{SparseShift}_R$ and show the following. 1. For every function $β: \mathbb{N}\to\mathbb{R}_+$ such that $β\in o(1)$, $N^β$-gap-$\mathrm{SparseShift}_\mathbb{Z}$ is also undecidable (where $N$ is the input length). 2. For $R=\mathbb{F}_p, \mathbb{Q}, \mathbb{R}$ or $\mathbb{Z}_q$ and for every $β>1$ the $β$-gap-$\mathrm{SparseShift}_R$ problem is $\mathrm{NP}$-hard.

cs.CC↗
arXiv2022

Robust Sylvester-Gallai type theorem for quadratic polynomials

In this work, we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if $\mathcal{Q}\subset \mathbb{C}[x_1.\ldots,x_n]$ is a finite set, $|\mathcal{Q}|=m$, of irreducible quadratic polynomials that satisfy the following condition: There is $δ>0$ such that for every $Q\in\mathcal{Q}$ there are at least $δm$ polynomials $P\in \mathcal{Q}$ such that whenever $Q$ and $P$ vanish then so does a third polynomial in $\mathcal{Q}\setminus\{Q,P\}$, then $\dim(\text{span}({\mathcal{Q}}))=\text{poly}(1/δ)$. The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of $O(1/δ)$ on the dimension (in the first work an upper bound of $O(1/δ^2)$ was given, which was improved to $O(1/δ)$ in the second work).

cs.CG↗
arXiv2022

Lower Bounds on Stabilizer Rank

The stabilizer rank of a quantum state $ψ$ is the minimal $r$ such that $\left| ψ\right \rangle = \sum_{j=1}^r c_j \left|φ_j \right\rangle$ for $c_j \in \mathbb{C}$ and stabilizer states $φ_j$. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the $n$-th tensor power of single-qubit magic states. We prove a lower bound of $Ω(n)$ on the stabilizer rank of such states, improving a previous lower bound of $Ω(\sqrt{n})$ of Bravyi, Smith and Smolin (arXiv:1506.01396). Further, we prove that for a sufficiently small constant $δ$, the stabilizer rank of any state which is $δ$-close to those states is $Ω(\sqrt{n}/\log n)$. This is the first non-trivial lower bound for approximate stabilizer rank. Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of $\mathbb{F}_2^n$, and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.

quant-ph↗
arXiv2022

Reed Solomon Codes Against Adversarial Insertions and Deletions

In this work, we study the performance of Reed--Solomon codes against adversarial insertion-deletion (insdel) errors. We prove that over fields of size $n^{O(k)}$ there are $[n,k]$ Reed-Solomon codes that can decode from $n-2k+1$ insdel errors and hence attain the half-Singleton bound. We also give a deterministic construction of such codes over much larger fields (of size $n^{k^{O(k)}}$). Nevertheless, for $k=O(\log n /\log\log n)$ our construction runs in polynomial time. For the special case $k=2$, which received a lot of attention in the literature, we construct an $[n,2]$ Reed-Solomon code over a field of size $O(n^4)$ that can decode from $n-3$ insdel errors. Earlier constructions required an exponential field size. Lastly, we prove that any such construction requires a field of size $Ω(n^3)$.

cs.IT↗
arXiv2022

Explicit and Efficient Constructions of linear Codes Against Adversarial Insertions and Deletions

In this work, we study linear error-correcting codes against adversarial insertion-deletion (insdel) errors, a topic that has recently gained a lot of attention. We construct linear codes over $\mathbb{F}_q$, for $q=\text{poly}(1/\varepsilon)$, that can efficiently decode from a $δ$ fraction of insdel errors and have rate $(1-4δ)/8-\varepsilon$. We also show that by allowing codes over $\mathbb{F}_{q^2}$ that are linear over $\mathbb{F}_q$, we can improve the rate to $(1-δ)/4-\varepsilon$ while not sacrificing efficiency. Using this latter result, we construct fully linear codes over $\mathbb{F}_2$ that can efficiently correct up to $δ< 1/54$ fraction of deletions and have rate $R = (1-54\cdot δ)/1216$. Cheng, Guruswami, Haeupler, and Li [CGHL21] constructed codes with (extremely small) rates bounded away from zero that can correct up to a $δ< 1/400$ fraction of insdel errors. They also posed the problem of constructing linear codes that get close to the half-Singleton bound (proved in [CGHL21]) over small fields. Thus, our results significantly improve their construction and get much closer to the bound.

cs.IT↗
arXiv2021

Hitting Sets and Reconstruction for Dense Orbits in $\text{VP}_e$ and $ΣΠΣ$ Circuits

In this paper we study polynomials in $\text{VP}_e$ (polynomial-sized formulas) and in $ΣΠΣ$ (polynomial-size depth-$3$ circuits) whose orbits, under the action of the affine group $\text{GL}_n^{\text{aff}}(\mathbb{F})$, are $\mathit{dense}$ in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. As $\text{VP}=\text{VNC}^2$, our results for $\text{VP}_e$ translate immediately to $\text{VP}$ with a quasipolynomial blow up in parameters. If any of our hitting or interpolating sets could be made $\mathit{robust}$ then this would immediately yield a hitting set for the superclass in which the relevant class is dense, and as a consequence also a lower bound for the superclass. Unfortunately, we also prove that the kind of constructions that we have found (which are defined in terms of $k$-independent polynomial maps) do not necessarily yield robust hitting sets.

cs.CC↗
arXiv2020

Sylvester-Gallai type theorems for quadratic polynomials

We prove Sylvester-Gallai type theorems for quadratic polynomials. Specifically, we prove that if a finite collection $\mathcal Q$, of irreducible polynomials of degree at most $2$, satisfy that for every two polynomials $Q_1,Q_2\in {\mathcal Q}$ there is a third polynomial $Q_3\in{\mathcal Q}$ so that whenever $Q_1$ and $Q_2$ vanish then also $Q_3$ vanishes, then the linear span of the polynomials in ${\mathcal Q}$ has dimension $O(1)$. We also prove a colored version of the theorem: If three finite sets of quadratic polynomials satisfy that for every two polynomials from distinct sets there is a polynomial in the third set satisfying the same vanishing condition then all polynomials are contained in an $O(1)$-dimensional space. This answers affirmatively two conjectures of Gupta [ECCC 2014] that were raised in the context of solving certain depth-$4$ polynomial identities. To obtain our main theorems we prove a new result classifying the possible ways that a quadratic polynomial $Q$ can vanish when two other quadratic polynomials vanish. Our proofs also require robust versions of a theorem of Edelstein and Kelly (that extends the Sylvester-Gallai theorem to colored sets).

math.CO↗
arXiv2020

Reed-Muller Codes: Theory and Algorithms

Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They are used in many areas of coding theory in both electrical engineering and computer science. Yet, many of their important properties are still under investigation. This paper covers some of the recent developments regarding the weight enumerator and the capacity-achieving properties of RM codes, as well as some of the algorithmic developments. In particular, the paper discusses the recent connections established between RM codes, thresholds of Boolean functions, polarization theory, hypercontractivity, and the techniques of approximating low weight codewords using lower degree polynomials. It then overviews some of the algorithms with performance guarantees, as well as some of the algorithms with state-of-the-art performances in practical regimes. Finally, the paper concludes with a few open problems.

cs.IT↗