arXiv subjects

Amik Raj Behera

Publications and source records attributed to Amik Raj Behera.

9 recordsLinked to original sources

arXiv2026

Low-Degree Testing Over Boolean Slices

We study low-degree testing for group-valued functions over a Boolean slice. Specifically given a degree parameter $d$ and oracle access to a function $f:\{0,1\}^n_{n/2}\to G$ where $\{0,1\}^n_k$ denotes the set of vectors in $\{0,1\}^n$ of Hamming weight $k$ and $G$ is an Abelian group, the low-degree testing problem asks us to distinguish the case where $f$ is a polynomial of degree at most $d$ (with coefficients from $G$) or is $\varepsilon$-far from the set of all such polynomials. Classical works in this area considered functions with domain $\mathbb{F}_q^n$ and range $\mathbb{F}_q$. More recent works have considered the setting where the domain is the Boolean cube [Bafna, Srinivasan, Sudan (Random Struct. Algorithms 2020), Amireddy, Srinivasan, Sudan (RANDOM 2023)], or when the domain is the slice (i.e., $\{0,1\}^n_{k}$) and the range is $\mathbb{F}_2$ [David, Dinur, Goldenberg, Kindler and Shinkar (SIAM J. Comput. 2017), Kalai, Lifshitz, Minzer and Ziegler (FOCS 2024)]. Each of the changes introduces new challenges in designing and analyzing low-degree tests and this happens again in our setting with domain being a slice and range is general. Our main theorem gives a test that makes $O_d(1)$ queries to $f$ and accepts degree-$d$ functions while rejecting functions that are $\varepsilon$-far with probability $Ω(\varepsilon)$. The central proof idea is to reduce this low-degree testing problem to the problem of low-degree testing on the cube. Specifically we show how to randomly embed the $n/2$-dimensional cube $\{0,1\}^{n/2}$ in the $n$-dimensional slice while nearly preserving the proximity of $f$ to the space of degree-$d$ polynomials on this cube. While the embedding is simple and natural, the analysis involves a careful induction with a novel use of a basis of degree-$d$ polynomials on slices (from a work of Anstee, Rónyai and Sali (Graphs and Combinatorics 2002)).

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arXiv2026

A Simple Algebraic Proof of the PCP Theorem

We give the simplest known algebraic proof of the PCP theorem, involving only ingredients like code concatenation, polynomial interpolation, and polynomial multiplication. Specifically, we prove that graph 3-coloring has a polynomial-sized proof that can be verified by a verifier tossing logarithmically many coins and querying a constant number of bits in the proof. In particular, our proof does not involve any PCP compositions; notably, it does not invoke the NP-completeness of any fixed problem, such as SAT or 3-coloring, in the construction of the verifier. The main innovation in our work is a clean, coding theoretic, way to encode univariate polynomials that allows us to implement ``low-degree testing'' using just a constant number of bits of queries. Insights from recent attempts to simplify the PCP proof by the authors (STOC 2026) and Goldreich (ECCC 2025) allow us to observe that low-degree was the key bottleneck in converting previous algebraic constructions of the PCP verifier into a constant query PCP. Thus, by overcoming this bottleneck, we get the full PCP verifier using elementary and self-contained steps. As concrete support for the claimed simplicity, we include the full pseudocode of the PCP verifier, assuming finite field arithmetic, and a full description of the completeness (aka ``honest'') prover, assuming multivariate polynomial arithmetic including interpolation and evaluation, that fit in about a page each.

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arXiv2026

Ideals, Macaulay Bases, and PCPs

All known proofs of the PCP theorem rely on multiple "composition" steps, where PCPs over large alphabets are turned into PCPs over much smaller alphabets at a (relatively) small price in the soundness error of the PCP. Algebraic proofs, starting with the work of Arora, Lund, Motwani, Sudan, and Szegedy use at least 2 such composition steps, whereas the "Gap amplification" proof of Dinur uses $Θ(\log n)$ such composition steps. In this work, we present the first PCP construction using just one composition step. The key ingredient, missing in previous work and finally supplied in this paper, is a basic PCP (of Proximity) of size $2^{n^ε}$, for any $ε> 0$, that makes $O_ε(1)$ queries. At the core of our new construction is a new class of alternatives to "sum-check" protocols. As used in past PCPs, these provide a method by which to verify that an $m$-variate degree $d$ polynomial $P$ evaluates to zero at every point of some set $S \subseteq \mathbb{F}_q^m$. Previous works had shown how to check this condition for sets of the form $S = H^m$ using $O(m)$ queries with alphabet $\mathbb{F}_q^d$ assuming $d \geq |H|$. Our work improves this basic protocol in two ways: First we extend it to broader classes of sets $S$ (ones closer to Hamming balls rather than cubes). Second, it reduces the number of queries from $O(m)$ to an absolute constant for the settings of $S$ we consider. Specifically when $S = (\{0,1\}^{m/c}_{\leq 1})^c$, we give such an alternate to the sum-check protocol with $O(1)$ queries with alphabet $\mathbb{F}_q^{O(c+d)}$, using proofs of size $q^{O(m^2/c)}$. Our new protocols use the notion of Macaulay bases to extend previously known protocols to these new settings with surprising ease. In doing so, they highlight why these notions from algebra may be of further use in complexity theory.

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arXiv2026

Separation Results for Constant-Depth and Multilinear Ideal Proof Systems

In this work, we establish separation theorems for several subsystems of the Ideal Proof System (IPS), an algebraic proof system introduced by Grochow and Pitassi (J. ACM, 2018). Separation theorems are well-studied in the context of classical complexity theory, Boolean circuit complexity, and algebraic complexity. In an important work of Forbes, Shpilka, Tzameret, and Wigderson (ToC, 2021), two proof techniques were introduced to prove lower bounds for subsystems of the IPS, namely the functional method and the multiples method. We use these techniques and obtain the following results. Hierarchy theorem for constant-depth IPS: Recently, Limaye, Srinivasan, and Tavenas (J. ACM 2025) proved a hierarchy theorem for constant-depth algebraic circuits. We adapt the result and prove a hierarchy theorem for constant-depth $\mathsf{IPS}$. We show that there is an unsatisfiable multilinear instance refutable by a depth-$Δ$ $\mathsf{IPS}$ such that any depth-($Δ/10)$ $\mathsf{IPS}$ refutation for it must have superpolynomial size. This result is proved by building on the multiples method. Separation theorems for multilinear IPS: In an influential work, Raz (ToC, 2006) unconditionally separated two algebraic complexity classes, namely multilinear $\mathsf{NC}^{1}$ from multilinear $\mathsf{NC}^{2}$. In this work, we prove a similar result for a well-studied fragment of multilinear-$\mathsf{IPS}$. Specifically, we present an unsatisfiable instance such that its functional refutation, i.e., the unique multilinear polynomial agreeing with the inverse of the polynomial over the Boolean cube, has a small multilinear-$\mathsf{NC}^{2}$ circuit. However, any multilinear-$\mathsf{NC}^{1}$ $\mathsf{IPS}$ refutation ($\mathsf{IPS}_{\mathsf{LIN}}$) for it must have superpolynomial size. This result is proved by building on the functional method.

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arXiv2025

Eigenvalue Bounds for Symmetric Markov Chains on Multislices With Applications

We consider random walks on ``balanced multislices'' of any ``grid'' that respects the ``symmetries'' of the grid, and show that a broad class of such walks are good spectral expanders. (A grid is a set of points of the form $\mathcal{S}^n$ for finite $\mathcal{S}$, and a balanced multi-slice is the subset that contains an equal number of coordinates taking every value in $\mathcal{S}$. A walk respects symmetries if the probability of going from $u = (u_1,\ldots,u_n)$ to $v = (v_1,\ldots,v_n)$ is invariant under simultaneous permutations of the coordinates of $u$ and $v$.) Our main theorem shows that, under some technical conditions, every such walk where a single step leads to an almost $\mathcal{O}(1)$-wise independent distribution on the next state, conditioned on the previous state, satisfies a non-trivially small singular value bound. We give two applications of our theorem to error-correcting codes: (1) We give an analog of the Ore-DeMillo-Lipton-Schwartz-Zippel lemma for polynomials, and junta-sums, over balanced multislices. (2) We also give a local list-correction algorithm for $d$-junta-sums mapping an arbitrary grid $\mathcal{S}^n$ to an Abelian group, correcting from a near-optimal $(\frac{1}{|\mathcal{S}|^{d}} - \varepsilon)$ fraction of errors for every $\varepsilon > 0$, where a $d$-junta-sum is a sum of (arbitrarily many) $d$-juntas (and a $d$-junta is a function that depends on only $d$ of the $n$ variables). Our proofs are obtained by exploring the representation theory of the symmetric group and merging it with some careful spectral analysis.

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arXiv2025

A Near-Optimal Polynomial Distance Lemma Over Boolean Slices

The celebrated Ore-DeMillo-Lipton-Schwartz-Zippel (ODLSZ) lemma asserts that n-variate non-zero polynomial functions of degree d over a field $\mathbb{F}$ are non-zero over any "grid" $S^n$ for finite subset $S \subseteq \mathbb{F}$, with probability at least $\max\{|S|^{-d/(|S|-1)},1-d/|S|\}$ over the choice of random point from the grid. In particular, over the Boolean cube ($S = \{0,1\} \subseteq \mathbb{F}$), the lemma asserts non-zero polynomials are non-zero with probability at least $2^{-d}$. In this work we extend the ODLSZ lemma optimally (up to lower-order terms) to "Boolean slices" i.e., points of Hamming weight exactly $k$. We show that non-zero polynomials on the slice are non-zero with probability $(t/n)^{d}(1 - o_{n}(1))$ where $t = \min\{k,n-k\}$ for every $d\leq k\leq (n-d)$. As with the ODLSZ lemma, our results extend to polynomials over Abelian groups. This bound is tight (upto the error term) as evidenced by degree d multilinear monomials. A particularly interesting case is the "balanced slice" ($k=n/2$) where our lemma asserts that non-zero polynomials are non-zero with roughly the same probability on the slice as on the whole cube. The behaviour of low-degree polynomials over Boolean slices has received much attention in recent years. However, the problem of proving a tight version of the ODLSZ lemma does not seem to have been considered before, except for a recent work of Amireddy, Behera, Paraashar, Srinivasan and Sudan (SODA 2025) who established a sub-optimal bound of approximately $((k/n)\cdot(1-(k/n)))^d$ using a proof similar to that of the standard ODLSZ lemma. While the statement of our result mimics that of the ODLSZ lemma, our proof is significantly more intricate and involves spectral reasoning which is employed to show that a natural way of embedding a copy of the Boolean cube inside a balanced Boolean slice is a good sampler.

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arXiv2025

New Bounds for the Ideal Proof System in Positive Characteristic

In this work, we prove upper and lower bounds over fields of positive characteristics for several fragments of the Ideal Proof System (IPS), an algebraic proof system introduced by Grochow and Pitassi (J. ACM 2018). Our results extend the works of Forbes, Shpilka, Tzameret, and Wigderson (Theory of Computing 2021) and also of Govindasamy, Hakoniemi, and Tzameret (FOCS 2022). These works primarily focused on proof systems over fields of characteristic $0$, and we are able to extend these results to positive characteristic. The question of proving general IPS lower bounds over positive characteristic is motivated by the important question of proving $AC^{0}[p]$-Frege lower bounds. This connection was observed by Grochow and Pitassi (J. ACM 2018). Additional motivation comes from recent developments in algebraic complexity theory due to Forbes (CCC 2024) who showed how to extend previous lower bounds over characteristic $0$ to positive characteristic. In our work, we adapt the functional lower bound method of Forbes et al. (Theory of Computing 2021) to prove exponential-size lower bounds for various subsystems of IPS. Additionally, we derive upper bounds for the instances presented above. We show that they have efficient constant-depth IPS refutations. We also show that constant-depth IPS can efficiently refute a general class of instances, namely all symmetric instances, thereby further uncovering the strength of these algebraic proofs in positive characteristic. Notably, our lower bounds hold for fields of arbitrary characteristic but require the field size to be $n^{ω(1)}$. In a concurrent work, Elbaz, Govindasamy, Lu, and Tzameret have shown lower bounds against restricted classes of IPS over finite fields of any size by considering different hard instances.

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arXiv2024

Low Degree Local Correction Over the Boolean Cube

In this work, we show that the class of multivariate degree-$d$ polynomials mapping $\{0,1\}^{n}$ to any Abelian group $G$ is locally correctable with $\widetilde{O}_{d}((\log n)^{d})$ queries for up to a fraction of errors approaching half the minimum distance of the underlying code. In particular, this result holds even for polynomials over the reals or the rationals, special cases that were previously not known. Further, we show that they are locally list correctable up to a fraction of errors approaching the minimum distance of the code. These results build on and extend the prior work of the authors [ABPSS24] (STOC 2024) who considered the case of linear polynomials and gave analogous results. Low-degree polynomials over the Boolean cube $\{0,1\}^{n}$ arise naturally in Boolean circuit complexity and learning theory, and our work furthers the study of their coding-theoretic properties. Extending the results of [ABPSS24] from linear to higher-degree polynomials involves several new challenges and handling them gives us further insights into properties of low-degree polynomials over the Boolean cube. For local correction, we construct a set of points in the Boolean cube that lie between two exponentially close parallel hyperplanes and is moreover an interpolating set for degree-$d$ polynomials. To show that the class of degree-$d$ polynomials is list decodable up to the minimum distance, we stitch together results on anti-concentration of low-degree polynomials, the Sunflower lemma, and the Footprint bound for counting common zeroes of polynomials. Analyzing the local list corrector of [ABPSS24] for higher degree polynomials involves understanding random restrictions of non-zero degree-$d$ polynomials on a Hamming slice. In particular, we show that a simple random restriction process for reducing the dimension of the Boolean cube is a suitably good sampler for Hamming slices.

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arXiv2024

Local Correction of Linear Functions over the Boolean Cube

We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain $\{0,1\}^n$ over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance $1/2$ and we give local-correction algorithms correcting up to nearly $1/4$-fraction errors making $\widetilde{\mathcal{O}}(\log n)$ queries. This query complexity is optimal up to $\mathrm{poly}(\log\log n)$ factors. We also give local list-correcting algorithms correcting $(1/2 - \varepsilon)$-fraction errors with $\widetilde{\mathcal{O}}_{\varepsilon}(\log n)$ queries. These results may be viewed as natural generalizations of the classical work of Goldreich and Levin whose work addresses the special case where the underlying group is $\mathbb{Z}_2$. By extending to the case where the underlying group is, say, the reals, we give the first non-trivial locally correctable codes (LCCs) over the reals (with query complexity being sublinear in the dimension (also known as message length)). The central challenge in constructing the local corrector is constructing "nearly balanced vectors" over $\{-1,1\}^n$ that span $1^n$ -- we show how to construct $\mathcal{O}(\log n)$ vectors that do so, with entries in each vector summing to $\pm1$. The challenge to the local-list-correction algorithms, given the local corrector, is principally combinatorial, i.e., in proving that the number of linear functions within any Hamming ball of radius $(1/2-\varepsilon)$ is $\mathcal{O}_{\varepsilon}(1)$. Getting this general result covering every Abelian group requires integrating a variety of known methods with some new combinatorial ingredients analyzing the structural properties of codewords that lie within small Hamming balls.

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