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1. 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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2. A Faster Solution to Smale's 17th Problem I: Real Binomial Systems

   https://doi.org/10.1145/3326229.3326267  

   Paouris, Grigoris; Phillipson, Kaitlyn; Rojas, J. Maurice (July 2019, ISSAC (International Symposium on Symbolic and Algebraic Computation) 2019)

   Suppose $$F:=(f_1,\ldots,f_n)$$ is a system of random $$n$$-variate polynomials with $$f_i$$ having degree $$\leq\!d_i$$ and the coefficient of $$x^{a_1}_1\cdots x^{a_n}_n$$ in $$f_i$$ being an independent complex Gaussian of mean $$0$$ and variance $$\frac{d_i!}{a_1!\cdots a_n!\left(d_i-\sum^n_{j=1}a_j \right)!}$$. Recent progress on Smale's 17$$\thth$$ Problem by Lairez --- building upon seminal work of Shub, Beltran, Pardo, B\"{u}rgisser, and Cucker --- has resulted in a deterministic algorithm that finds a single (complex) approximate root of $$F$$ using just $$N^{O(1)}$$ arithmetic operations on average, where $$N\!:=\!\sum^n_{i=1}\frac{(n+d_i)!}{n!d_i!}$$ ($$=n(n+\max_i d_i)^{O(\min\{n,\max_i d_i)\}}$$) is the maximum possible total number of monomial terms for such an $$F$$. However, can one go faster when the number of terms is smaller, and we restrict to real coefficient and real roots? And can one still maintain average-case polynomial-time with more general probability measures? We show the answer is yes when $$F$$ is instead a binomial system --- a case whose numerical solution is a key step in polyhedral homotopy algorithms for solving arbitrary polynomial systems. We give a deterministic algorithm that finds a real approximate root (or correctly decides there are none) using just $$O(n^3\log^2(n\max_i d_i))$$ arithmetic operations on average. Furthermore, our approach allows Gaussians with arbitrary variance. We also discuss briefly the obstructions to maintaining average-case time polynomial in $$n\log \max_i d_i$$ when $$F$$ has more terms. 

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3. Computing Higher Polynomial Discriminants

   https://doi.org/10.1145/3452143.3465543  

   Kaltofen, Erich L. (July 2021, ISSAC '21: Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation)

   null (Ed.)

   In https://arxiv.org/abs/1609.00840, D. Wang and J. Yang in 2016 have posed the problem how to compute the ``third'' discriminant of a polynomial f(X) = (X-A_1) ... (X-A_n), delta_3(f) = product_{1 <= i < j < k < l <= n} ( (A_i + A_j - A_k - A_l) (A_i - A_j + A_k - A_l) (A_i - A_j - A_k + A_l) ), from the coefficients of f; note that delta_3 is a symmetric polynomial in the A_i. For complex roots, delta_3(f) = 0 if the mid-point (average) of 2 roots is equal the mid-point of another 2 roots. Iterated resultant computations yield the square of the third discriminant. We apply a symbolic homotopy by Kaltofen and Trager [JSC, vol. 9, nr. 3, pp. 301--320 (1990)] to compute its squareroot. Our algorithm use polynomially many coefficient field operations. 

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4. Homological mirror symmetry for higher-dimensional pairs of pants

   https://doi.org/10.1112/S0010437X20007150  

   Lekili, Yankı; Polishchuk, Alexander (July 2020, Compositio Mathematica)

   Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $$\mathbb{CP}^{n}$$ , for $$k\geqslant n$$ , with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $$\mathbb{C}P^{n}$$ ( $$n$$ -dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $$x_{1}x_{2}\ldots x_{n+1}=0$$ . By localizing, we deduce that the (fully) wrapped Fukaya category of the $$n$$ -dimensional pair of pants is equivalent to the derived category of $$x_{1}x_{2}\ldots x_{n+1}=0$$ . We also prove similar equivalences for finite abelian covers of the $$n$$ -dimensional pair of pants. 

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5. NP-Hardness of Reed-Solomon Decoding and the Prouhet-Tarry-Escott Problem

   https://doi.org/10.1109/FOCS.2016.86  

   Gandikota, Venkata; Ghazi, Badih; Grigorescu, Elena (October 2016, FOCS)

   Establishing the complexity of {\em Bounded Distance Decoding} for Reed-Solomon codes is a fundamental open problem in coding theory, explicitly asked by Guruswami and Vardy (IEEE Trans. Inf. Theory, 2005). The problem is motivated by the large current gap between the regime when it is NP-hard, and the regime when it is efficiently solvable (i.e., the Johnson radius). We show the first NP-hardness results for asymptotically smaller decoding radii than the maximum likelihood decoding radius of Guruswami and Vardy. Specifically, for Reed-Solomon codes of length $$N$$ and dimension $K=O(N)$, we show that it is NP-hard to decode more than $$ N-K- c\frac{\log N}{\log\log N}$$ errors (with $c>0$ an absolute constant). Moreover, we show that the problem is NP-hard under quasipolynomial-time reductions for an error amount $$> N-K- c\log{N}$$ (with $c>0$$ an absolute constant). An alternative natural reformulation of the Bounded Distance Decoding problem for Reed-Solomon codes is as a {\em Polynomial Reconstruction} problem. In this view, our results show that it is NP-hard to decide whether there exists a degree $$K$$ polynomial passing through $$K+ c\frac{\log N}{\log\log N}$$ points from a given set of points $$(a_1, b_1), (a_2, b_2)\ldots, (a_N, b_N)$$. Furthermore, it is NP-hard under quasipolynomial-time reductions to decide whether there is a degree $$K$$ polynomial passing through $$K+c\log{N}$$ many points. These results follow from the NP-hardness of a generalization of the classical Subset Sum problem to higher moments, called {\em Moments Subset Sum}, which has been a known open problem, and which may be of independent interest. We further reveal a strong connection with the well-studied Prouhet-Tarry-Escott problem in Number Theory, which turns out to capture a main barrier in extending our techniques. We believe the Prouhet-Tarry-Escott problem deserves further study in the theoretical computer science community. 

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