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1. Reconstruction algorithms for low-rank tensors and depth-3 multilinear circuits

   https://doi.org/10.1145/3406325.3451096  

   Bhargava, Vishwas; Saraf, Shubhangi; Volkovich, Ilya (June 2021, STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing)

   null (Ed.)

   We give new and efficient black-box reconstruction algorithms for some classes of depth-3 arithmetic circuits. As a consequence, we obtain the first efficient algorithm for computing the tensor rank and for finding the optimal tensor decomposition as a sum of rank-one tensors when then input is a constant-rank tensor. More specifically, we provide efficient learning algorithms that run in randomized polynomial time over general fields and in deterministic polynomial time over and for the following classes: 1) Set-multilinear depth-3 circuits of constant top fan-in ((k) circuits). As a consequence of our algorithm, we obtain the first polynomial time algorithm for tensor rank computation and optimal tensor decomposition of constant-rank tensors. This result holds for d dimensional tensors for any d, but is interesting even for d=3. 2) Sums of powers of constantly many linear forms ((k) circuits). As a consequence we obtain the first polynomial-time algorithm for tensor rank computation and optimal tensor decomposition of constant-rank symmetric tensors. 3) Multilinear depth-3 circuits of constant top fan-in (multilinear (k) circuits). Our algorithm works over all fields of characteristic 0 or large enough characteristic. Prior to our work the only efficient algorithms known were over polynomially-sized finite fields (see. Karnin-Shpilka 09’). Prior to our work, the only polynomial-time or even subexponential-time algorithms known (deterministic or randomized) for subclasses of (k) circuits that also work over large/infinite fields were for the setting when the top fan-in k is at most 2 (see Sinha 16’ and Sinha 20’). 

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2. LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER

   https://doi.org/10.1112/S0025579319000032  

   Bienvenu, Pierre-Yves; Lê, Thái Hoàng (January 2019, Mathematika)

   We examine correlations of the Möbius function over $$\mathbb{F}_{q}[t]$$ with linear or quadratic phases, that is, averages of the form 1 $$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f0$$ if $$Q$$ is linear and $$O(q^{-n^{c}})$$ for some absolute constant $c>0$ if $$Q$$ is quadratic. The latter bound may be reduced to $$O(q^{-c^{\prime }n})$$ for some $$c^{\prime }>0$$ when $Q(f)$ is a linear form in the coefficients of $$f^{2}$$ , that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem. 

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3. New Pseudorandom Generators and Correlation Bounds Using Extractors

   https://doi.org/10.4230/LIPIcs.ITCS.2025.68  

   Kumar, Vinayak M (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   We establish new correlation bounds and pseudorandom generators for a collection of computation models. These models are all natural generalization of structured low-degree polynomials that we did not have correlation bounds for before. In particular: - We construct a PRG for width-2 poly(n)-length branching programs which read d bits at a time with seed length 2^O(√{log n}) ⋅ d²log²(1/ε). This comes quadratically close to optimal dependence in d and log(1/ε). Improving the dependence on n would imply nontrivial PRGs for log n-degree 𝔽₂-polynomials. The previous PRG by Bogdanov, Dvir, Verbin, and Yehudayoff had an exponentially worse dependence on d with seed length of O(dlog n + d2^dlog(1/ε)). - We provide the first nontrivial (and nearly optimal) correlation bounds and PRGs against size-n^Ω(log n) AC⁰ circuits with either n^{.99} SYM gates (computing an arbitrary symmetric function) or n^{.49} THR gates (computing an arbitrary linear threshold function). This is a generalization of sparse 𝔽₂-polynomials, which can be simulated by an AC⁰ circuit with one parity gate at the top. Previous work of Servedio and Tan only handled n^{.49} SYM gates or n^{.24} THR gates, and previous work of Lovett and Srinivasan only handled polynomial-size circuits. - We give exponentially small correlation bounds against degree-n^O(1) 𝔽₂-polynomials which are set-multilinear over some arbitrary partition of the input into n^{1-O(1)} parts (noting that at n parts, we recover all low degree polynomials). This vastly generalizes correlation bounds against degree-d polynomials which are set-multilinear over a fixed partition into d blocks, which were established by Bhrushundi, Harsha, Hatami, Kopparty, and Kumar. The common technique behind all of these results is to fortify a hard function with the right type of extractor to obtain stronger correlation bounds for more general models of computation. Although this technique has been used in previous work, they rely on the model simplifying drastically under random restrictions. We view our results as a proof of concept that such fortification can be done even for classes that do not enjoy such behavior. 

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4. On 2-superirreducible polynomials over finite fields

   https://doi.org/10.1016/j.indag.2024.08.005  

   Bober, Jonathan W; Du, Lara; Fretwell, Daniel; Kopp, Gene S; Wooley, Trevor D (May 2025, Indagationes Mathematicae)

   We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity. 

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5. Leibniz International Proceedings in Informatics (LIPIcs):38th Computational Complexity Conference (CCC 2023)

   https://doi.org/10.4230/LIPIcs.CCC.2023.4  

   Galesi, Nicola; Grochow, Joshua A.; Pitassi, Toniann; She, Adrian (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Ta-Shma, Amnon (Ed.)

   The Tensor Isomorphism problem (TI) has recently emerged as having connections to multiple areas of research within complexity and beyond, but the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, TI (or rather, proving that two tensors are non-isomorphic) lends itself very naturally to algebraic and semi-algebraic proof systems, such as the Polynomial Calculus (PC) and Sum of Squares (SoS). For its combinatorial cousin Graph Isomorphism, essentially optimal lower bounds are known for approaches based on PC and SoS (Berkholz & Grohe, SODA '17). Our main results are an Ω(n) lower bound on PC degree or SoS degree for Tensor Isomorphism, and a nontrivial upper bound for testing isomorphism of tensors of bounded rank. We also show that PC cannot perform basic linear algebra in sub-linear degree, such as comparing the rank of two matrices (which is essentially the same as 2-TI), or deriving BA=I from AB=I. As linear algebra is a key tool for understanding tensors, we introduce a strictly stronger proof system, PC-Inv, which allows as derivation rules all substitution instances of the implication AB=I → BA=I. We conjecture that even PC-Inv cannot solve TI in polynomial time either, but leave open getting lower bounds on PC-Inv for any system of equations, let alone those for TI. We also highlight many other open questions about proof complexity approaches to TI. 

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