arXiv subjects

Andrei Mandelshtam

Publications and source records attributed to Andrei Mandelshtam.

5 recordsLinked to original sources

arXiv2025

Low-degree minimal generating sets of polynomial ideals

We obtain tight bounds for the minimal number of generators of an ideal with bounded-degree generators in a polynomial ring $K[X_1,\dots,X_n],$ as well as a sharp quantification of the maximum possible size of a minimal generating set of bounded degree. Our bounds are sharp for all fields of size greater than the degree. Moreover, we provide explicit constructions reaching the tightness constraints for all fields of characteristic 0, and for all sufficiently large fields in the one- and two-variable case. Additionally, we fully solve the one-variable case, and conjecture the asymptotics in the multivariate case, for all finite fields.

math.AC↗

arXiv2024

Subconvexity Implies Effective Quantum Unique Ergodicity for Hecke-Maa{\ss} Cusp Forms on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R})$

It is a folklore result in arithmetic quantum chaos that quantum unique ergodicity on the modular surface with an effective rate of convergence follows from subconvex bounds for certain triple product $L$-functions. The physical space manifestation of this result, namely the equidistribution of mass of Hecke-Maass cusp forms, was proven to follow from subconvexity by Watson, whereas the phase space manifestation of quantum unique ergodicity has only previously appeared in the literature for Eisenstein series via work of Jakobson. We detail the analogous phase space result for Hecke-Maass cusp forms. The proof relies on the Watson-Ichino triple product formula together with a careful analysis of certain archimedean integrals of Whittaker functions.

math.NT↗

arXiv2022

On fractal patterns in Ulam words

Ulam words are binary words defined recursively as follows: the length-$1$ Ulam words are $0$ and $1$, and a binary word of length $n$ is Ulam if and only if it is expressible uniquely as a concatenation of two shorter, distinct Ulam words. We discover, fully describe, and prove a surprisingly rich structure already in the set of Ulam words containing exactly two $1$'s. In particular, this leads to a complete description of such words and a logarithmic-time algorithm to determine whether a binary word with two $1$'s is Ulam. Along the way, we uncover delicate parity and biperiodicity properties, as well as sharp bounds on the number of $0$'s outside the two $1$'s. We also show that sets of Ulam words indexed by the number $y$ of $0$'s between the two $1$'s have intricate tensor-based hierarchical structures determined by the arithmetic properties of $y$. This allows us to construct an infinite family of self-similar Ulam-word-based fractals indexed by the set of $2$-adic integers, containing the outward Sierpinski gasket as a special case.

math.CO↗

arXiv2022

Symbol Length in Brauer Groups of Elliptic Curves

Let $\ell$ be an odd prime, and let $K$ be a field of characteristic not $2,3,$ or $\ell$ containing a primitive $\ell$-th root of unity. For an elliptic curve $E$ over $K$, we consider the standard Galois representation $$ρ_{E,\ell}: \text{Gal}(\overline{K}/K) \rightarrow \text{GL}_2(\mathbb{F}_{\ell}),$$ and denote the fixed field of its kernel by $L$. Recently, the last author gave an algorithm to compute elements in the Brauer group explicitly, deducing an upper bound of $2(\ell+1)(\ell-1)$ on the symbol length in $\mathbin{_{\ell}\text{Br}(E)} / \mathbin{_{\ell}\text{Br}(K)}$. More precisely, the symbol length is bounded above by $2[L:K]$. We improve this bound to $[L:K]-1$ if $\ell \nmid [L:K]$. Under the additional assumption that $\text{Gal}(L/K)$ contains an element of order $d > 1$, we further reduce it to $(1-\frac{1}{d})[L:K]$. In particular, these bounds hold for all CM elliptic curves, in which case we deduce a general upper bound of $\ell + 1$. We provide an algorithm implemented in SageMath to compute these symbols explicitly over number fields.

math.NT↗

arXiv2020

Sums of binomial coefficients evaluated at $α\in \overline{\mathbb Q},$ and applications

The additive monoid $R_+(x)$ is defined as the set of all nonnegative integer linear combinations of binomial coefficients $\binom{x}{n}$ for $n \in \mathbb Z_+$. This paper is concerned with the inquiry into the structure of $R_+(α)$ for complex numbers $α.$ Particularly interesting is the case of algebraic $α$ which are not non-negative integers. This question is motivated by the study of functors between Deligne categories $\textrm{Rep}(S_t)$ (and also $\textrm{Rep}(\textrm{GL}_t)$) for $t \in \mathbb C\backslash \mathbb Z_+$. We prove that this object is a ring if and only if $α$ is an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by $α,$ i.e. all elements of $\mathcal O_{\mathbb Q(α)},$ are also contained in this ring. We also give two explicit representations of $R_+(α)$ for both algebraic integers and general algebraic numbers $α.$ One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. We show how these results work in the context of the study of symmetric monoidal functors between Deligne categories in positive characteristic. Moreover, this leads to a particularly simple description of $R_+(α)$ for both quadratic algebraic numbers and roots of unity.

math.RT↗