Adversarial Wronskians: A proof of the Kung--Traub conjecture

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1. Related Broader context Not relevant

   An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

   Matthew Kwan, Lisa Sauermann · 2019 · Combinatorics · cited 7 times · AI summary available

   The quadratic Littlewood-Offord paper studies concentration and algebraic structure of random quadratic polynomials, rather than limits on adaptive root-finding evaluations.

   Abstract

   Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of $f$ can be as large as about $1/\sqrt{n}$, but still poorly understood is the "inverse" question of characterising the algebraic and arithmetic features $f$ must have if it has point probabilities comparable to this bound. In this paper we prove some results of an algebraic flavour, showing that if $f$ has point probabilities much larger…

2. Related Broader context Not relevant

   Certified homotopy tracking using the Krawczyk method

   Timothy Duff, Kisun Lee · 2024 · Numerical Analysis · cited 14 times · AI summary available

   Certified homotopy tracking uses interval arithmetic to verify solution paths of polynomial systems, not to bound the convergence order of root-finding methods.

   Abstract

   We revisit the problem of certifying the correctness of approximate solution paths computed by numerical homotopy continuation methods. We propose a conceptually simple approach based on a parametric variant of the Krawczyk method from interval arithmetic. Unlike most previous methods for certified path-tracking, our approach is applicable in the general setting of parameter homotopies commonly used to solve polynomial systems of equations. We also describe a novel preconditioning strategy and give theoretical correctness and termination results. Experiments using a preliminary implementation of the method indicate that our approach is competitive with specialized methods appearing previousl…

3. Related Broader context Not relevant

   Weak Poincaré Inequalities via Approximate Stochastic Localization: Application to Sampling the Sherrington-Kirkpatrick Model

   Ewan Davies, Holden Lee, Juspreet Singh Sandhu et al. · 2026 · Probability · cited 3 times · AI summary available

   The stochastic-localization paper proves a weak Poincaré inequality for the Sherrington-Kirkpatrick model and applies it to sampling, not root finding.

   Abstract

   We develop a new method for proving a weak functional inequality by first proving it for a sufficiently regular sequence of distributions approximating the stochastic localization (SL) process, and then transferring it to the desired distribution via regularity of the SL process and conductance arguments. We use this strategy to prove a weak Poincaré inequality (WPI) holds for the Gibbs measure of the Sherrington-Kirkpatrick model when $β< \frac{1}{2}$. A prior result of the authors [arXiv:2605.03718, 2026] proves the ASL process for the Sherrington-Kirkpatrick model satisfies the required regularity conditions. A consequence of the WPI is that a much simpler algorithm -- Glauber dynamics wi…

4. Related Broader context Not relevant

   Oblivious Sketching of High-Degree Polynomial Kernels

   Michael Kapralov, Rasmus Pagh, Ameya Velingker et al. · 2019 · Data Structures and Algorithms · AI summary available

   The sketching paper develops algorithms for speeding up high-degree polynomial kernel computations, rather than studying root-finding convergence limits.

   Abstract

   Kernel methods are fundamental tools in machine learning that allow detection of non-linear dependencies between data without explicitly constructing feature vectors in high dimensional spaces. A major disadvantage of kernel methods is their poor scalability: primitives such as kernel PCA or kernel ridge regression generally take prohibitively large quadratic space and (at least) quadratic time, as kernel matrices are usually dense. Some methods for speeding up kernel linear algebra are known, but they all invariably take time exponential in either the dimension of the input point set (e.g., fast multipole methods suffer from the curse of dimensionality) or in the degree of the kernel functi…

5. Related Broader context Not relevant

   Potential Hessian Ascent: The Sherrington-Kirkpatrick Model

   David Jekel, Juspreet Singh Sandhu, Jonathan Shi · 2024 · Probability · preprint, not peer reviewed · cited 8 times · AI summary available

   Potential Hessian Ascent develops a spectral algorithm for optimizing random quadratic objectives on the hypercube, not for finding analytic function zeros.

   Abstract

   We provide the first iterative spectral algorithm to find near-optima of a random quadratic objective over the discrete hypercube. The algorithm is a randomized Hessian ascent in the solid cube, where we modify the objective by subtracting a specific instance-independent potential function [Chen et al., Communications on Pure and Applied Mathematics, 76(7), 2023]. This extends Subag's algorithmic program of Hessian ascent from the sphere [Subag, Communications on Pure and Applied Mathematics, 74(5), 2021] to the more complex geometry of the cube. Utilizing tools from free probability theory, we construct an approximate projector into the top-eigenspaces of the Hessian with well-behaved diago…

6. Related Broader context Not relevant

   The Hamel-Nadirashvili conjecture for the Fisher-KPP equation

   Hongjun Guo, Kelei Wang, Maolin Zhou · 2026 · Analysis of PDEs · AI summary available

   The Fisher-KPP paper classifies entire solutions of a reaction-diffusion equation, rather than analyzing evaluation limits in root-finding methods.

   In short

   The paper proves that all bounded entire Fisher–KPP solutions with concave reaction terms are local limits of finite mixtures of planar fronts, optionally including a homogeneous solution.

7. Related Broader context Not relevant

   Matrix anti-concentration inequalities with applications

   Zipei Nie · 2021 · Probability · cited 22 times · AI summary available

   The matrix anti-concentration paper bounds minimum singular values of random matrices, not the convergence order of deterministic root-finding methods.

   Abstract

   We provide a polynomial lower bound on the minimum singular value of an $m\times m$ random matrix $M$ with jointly Gaussian entries, under a polynomial bound on the matrix norm and a global small-ball probability bound $$\inf_{x,y\in S^{m-1}}\mathbb{P}\left(\left|x^* M y\right|>m^{-O(1)}\right)\ge \frac{1}{2}.$$ With the additional assumption that $M$ is self-adjoint, the global small-ball probability bound can be replaced with a weaker version. We establish two matrix anti-concentration inequalities, which lower bound the minimum singular values of the sum of independent positive semidefinite self-adjoint matrices and the linear combination of independent random matrices with independent Ga…

8. Related Broader context Not relevant

   Fast Multivariate Multipoint Evaluation Over All Finite Fields

   Vishwas Bhargava, Sumanta Ghosh, Zeyu Guo et al. · 2022 · Data Structures and Algorithms · cited 21 times · AI summary available

   Fast Multivariate Multipoint Evaluation develops algorithms to evaluate polynomials at many finite-field points, rather than to find analytic function zeros.

   Abstract

   Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field $\mathbb{F}$ that outputs the evaluations of an $m$-variate polynomial of degree less than $d$ in each variable at $N$ points in time \[ (d^m+N)^{1+o(1)}\cdot\poly(m,d,\log|\mathbb{F}|) \] for all $m\in\N$ and all sufficiently large $d\in\mathbb{N}$. A previous work of Kedlaya and Umans (FOCS 2008, SICOMP 2011) achieved the same time complexity when the number of variables $m$ is at most $d^{o(1)}$ and…