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1. Dimension-free discretizations of the uniform norm by small product sets

   https://doi.org/10.1007/s00222-024-01306-9  

   Becker, Lars; Klein, Ohad; Slote, Joseph; Volberg, Alexander; Zhang, Haonan (December 2024, Inventiones mathematicae)

   Abstract Let$$f$$ $f$be an analytic polynomial of degree at most$$K-1$$ $K-1$. A classical inequality of Bernstein compares the supremum norm of$$f$$ $f$over the unit circle to its supremum norm over the sampling set of the$$K$$ $K$-th roots of unity. Many extensions of this inequality exist, often understood under the umbrella of Marcinkiewicz–Zygmund-type inequalities for$$L^{p},1\le p\leq \infty $$ $L^p,1\le p\le \infty$norms. We study dimension-free extensions of these discretization inequalities in the high-dimension regime, where existing results construct sampling sets with cardinality growing with the total degree of the polynomial. In this work we show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of$$\deg (f)$$ $deg\left(f\right)$and is instead governed by the maximumindividualdegree of$$f$$ $f$;i.e., the largest degree of$$f$$ $f$when viewed as a univariate polynomial in any coordinate. For example, we find that for$$n$$ $n$-variate analytic polynomials$$f$$ $f$of degree at most$$d$$ $d$and individual degree at most$$K-1$$ $K-1$,$$\|f\|_{L^{\infty }(\mathbf{D}^{n})}\leq C(X)^{d}\|f\|_{L^{\infty }(X^{n})}$$ ${\parallel f\parallel }_{L^{\infty }\left(D^n\right)}\le C{\left(X\right)}^d{\parallel f\parallel }_{L^{\infty }\left(X^n\right)}$for any fixed$$X$$ $X$in the unit disc$$\mathbf{D}$$ $D$with$$|X|=K$$ $\left|X\right|=K$. The dependence on$$d$$ $d$in the constant is tight for such small sampling sets, which arise naturally for example when studying polynomials of bounded degree coming from functions on products of cyclic groups. As an application we obtain a proof of the cyclic group Bohnenblust–Hille inequality with an explicit constant$$\mathcal{O}(\log K)^{2d}$$ $O{(logK)}^{2d}$. 

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2. Hilbert Functions and Low-Degree Randomness Extractors

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.41  

   Golovnev, Alexander; Guo, Zeyu; Hatami, Pooya; Nagargoje, Satyajeet; Yan, Chao (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Kumar, Amit; Ron-Zewi, Noga (Ed.)

   For S ⊆ 𝔽ⁿ, consider the linear space of restrictions of degree-d polynomials to S. The Hilbert function of S, denoted h_S(d,𝔽), is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets S of arbitrary finite grids in 𝔽ⁿ with a fixed size |S|. We achieve this by proving that this value coincides with a combinatorial quantity, namely the smallest number of low Hamming weight points in a down-closed set of size |S|. Understanding the smallest values of Hilbert functions is closely related to the study of degree-d closure of sets, a notion introduced by Nie and Wang (Journal of Combinatorial Theory, Series A, 2015). We use bounds on the Hilbert function to obtain a tight bound on the size of degree-d closures of subsets of 𝔽_qⁿ, which answers a question posed by Doron, Ta-Shma, and Tell (Computational Complexity, 2022). We use the bounds on the Hilbert function and degree-d closure of sets to prove that a random low-degree polynomial is an extractor for samplable randomness sources. Most notably, we prove the existence of low-degree extractors and dispersers for sources generated by constant-degree polynomials and polynomial-size circuits. Until recently, even the existence of arbitrary deterministic extractors for such sources was not known. 

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3. Low Degree Local Correction Over the Boolean Cube

   Amireddy, Prashanth; Behera, Amik Raj; Paraashar, Manaswi; Srinivasan, Srikanth; Sudan, Madhu (January 2025, Society for Industrial and Applied Mathematics)

   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 Õd((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 Amireddy, Behera, Paraashar, Srinivasan, and Sudan [1] (STOC 2024) who considered the case of linear polynomials (d = 1) 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 [1] from linear polynomials 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 [1] 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. Thus our exploration unearths several new techniques that are useful in understanding the combinatorial structure of low-degree polynomials over {0, 1}n. 

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4. Low Degree Local Correction Over the Boolean Cube

   Amireddy, Prashanth; Behera, Amik Raj; Paraashar, Manaswi; Srinivasan, Srikanth; Sudan, Madhu (January 2025, Society for Industrial and Applied Mathematics)

   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 Õd((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 Amireddy, Behera, Paraashar, Srinivasan, and Sudan [1] (STOC 2024) who considered the case of linear polynomials (d = 1) 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 [1] from linear polynomials 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 [1] 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. Thus our exploration unearths several new techniques that are useful in understanding the combinatorial structure of low-degree polynomials over {0, 1}n. 

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5. Finite Matrix Multiplication Algorithms from Infinite Groups

   https://doi.org/10.4230/LIPIcs.ITCS.2025.18  

   Blasiak, Jonah; Cohn, Henry; Grochow, Joshua A; Pratt, Kevin; Umans, Chris (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group G satisfying a simple combinatorial condition (the Triple Product Property). The complexity of such an algorithm then depends on the representation theory of G. In this paper we extend the group-theoretic framework to the setting of infinite groups. In particular, this allows us to obtain constructions in Lie groups, with favorable parameters, that are provably impossible in finite groups of Lie type (Blasiak, Cohn, Grochow, Pratt, and Umans, ITCS '23). Previously the Lie group setting was investigated purely as an analogue of the finite group case; a key contribution in this paper is a fully developed framework for obtaining bona fide matrix multiplication algorithms directly from Lie group constructions. As part of this framework, we introduce "separating functions" as a necessary new design component, and show that when the underlying group is G = GL_n, these functions are polynomials with their degree being the key parameter. In particular, we show that a construction with "half-dimensional" subgroups and optimal degree would imply ω = 2. We then build up machinery that reduces the problem of constructing optimal-degree separating polynomials to the problem of constructing a single polynomial (and a corresponding set of group elements) in a ring of invariant polynomials determined by two out of the three subgroups that satisfy the Triple Product Property. This machinery combines border rank with the Lie algebras associated with the Lie subgroups in a critical way. We give several constructions illustrating the main components of the new framework, culminating in a construction in a special unitary group that achieves separating polynomials of optimal degree, meeting one of the key challenges. The subgroups in this construction have dimension approaching half the ambient dimension, but (just barely) too slowly. We argue that features of the classical Lie groups make it unlikely that constructions in these particular groups could produce nontrivial bounds on ω unless they prove ω = 2. One way to get ω = 2 via our new framework would be to lift our existing construction from the special unitary group to GL_n, and improve the dimension of the subgroups from (dim G)/2 - Θ(n) to (dim G)/2 - o(n). 

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