arXiv subjects

Benjamin Baily

Publications and source records attributed to Benjamin Baily.

8 recordsLinked to original sources

arXiv2026

Homogeneous ideals with minimal singularity thresholds

Let $(\mathcal{O}_n, \mathfrak{m})$ denote the ring of germs of holomorphic functions $\mathbb{C}^n\to \mathbb{C}$, and let $I\subseteq \mathcal{O}_n$ be an $\mathfrak{m}$-primary ideal. Demailly and Pham showed that $\mathrm{lct}(I) \geq \frac{1}{e_1(I)} + \dots + \frac{e_{n-1}(I)}{e_n(I)}$, where $e_j(I)$ is the mixed multiplicity $e(I,\dots, I, \mathfrak{m},\dots, \mathfrak{m})$, with $I$ repeated $j$ times and $\mathfrak{m}$ repeated $n-j$ times. We generalize the lower bound to the case of an arbitrary ideal of an excellent regular local (or standard-graded) ring of equal characteristic, with $\mathrm{lct}(I)$ replaced by the $F$-threshold $c^{\mathfrak{m}}(I)$ in positive characteristic. Our main result is a classification of homogeneous ideals in polynomial rings for which the lower bound is attained, resolving a conjecture of Bivi\`a-Ausina in the graded case.

math.AC↗

arXiv2026

Extremal F-thresholds in regular local rings

Let $(R, \mathfrak{m})$ be a regular local ring of characteristic $p > 0$. Among all proper ideals $\mathfrak{a}\subseteq R$ with a fixed order of vanishing $\text{ord}_{\mathfrak{m}}(\mathfrak{a})$, we classify the ideals for which the $F$-threshold $\text{ft}^{\mathfrak{m}}(\mathfrak{a})$ is minimal.

math.AC↗

arXiv2025

On lower bounds for the F-pure threshold of equigenerated ideals

Let $k$ be a field of positive characteristic and $R = k[x_0,\dots, x_n]$. We consider ideals $I\subseteq R$ generated by homogeneous polynomials of degree $d$. Takagi and Watanabe proved that $\mathrm{fpt}(I)\geq \mathrm{height}(I)/d$; we classify ideals $I$ for which equality is attained. Inspired by a result of de Fernex, Ein, and Musta\c{t}\u{a}, we give a lower bound on $\mathrm{fpt}(I)$ in terms of the height of $\tau(I^{\mathrm{fpt}(I)})$.

math.AC↗

arXiv2025

On deformation of perfectoid purity in Gorenstein domains

If $(R,\mathfrak{m})$ is a complete local ring of mixed characteristic $(0,p)$ and $R/pR$ is an $F$-pure Gorenstein domain, we find a sufficient condition for $R$ to be perfectoid pure. This condition is related to the Cohen-Macaulayness of the absolute integral closures of Gorenstein local domains of mixed characteristic which are not necessarily excellent. Along the way, we show that the problem of lifting $F$-purity of $R/pR$ to perfectoid purity of $R$ is equivalent to a similar deformation problem for the splinter property.

math.AC↗

arXiv2024

Strength is bounded linearly by Birch rank

Let $f$ be a homogeneous polynomial over a field. For many fields, including number fields and function fields, we prove that the strength of $f$ is bounded above by a constant multiple of the Birch rank of $f.$ The constant depends only on the degree of $f$ and the absolute transcendence degree of the field. This is the first linear bound obtained for forms of degree greater than three, partially resolving a conjecture of Adiprasito, Kazhdan and Ziegler. Our result has applications for the Hardy-Littlewood circle method. The circle method yields an asymptotic formula for counting integral zeros of (collections of) homogeneous polynomials, provided the Birch rank is sufficiently large -- a natural geometric condition. Our main theorem implies that these formulas hold even if we only assume a similar lower bound on the strength of the (collection of) homogeneous polynomials -- an arithmetic condition which is a priori weaker. This answers questions of Cook-Magyar and Skinner, and also yields a new proof of a seminal result of Schmidt as a consequence of Birch's earlier work. Over finite fields we obtain a quasi-linear bound for partition rank of tensors in terms of analytic rank, improving Moshkovitz-Zhu's state of the art bound.

math.NT↗

arXiv2021

Irreducibility over the Max-Min Semiring

For sets $A, B\subset \mathbb N$, their sumset is $A + B := \{a+b: a\in A, b\in B\}$. If we cannot write a set $C$ as $C = A+B$ with $|A|, |B|\geq 2$, then we say that $C$ is $\textit{irreducible}$. The question of whether a given set $C$ is irreducible arises naturally in additive combinatorics. Equivalently, we can formulate this question as one about the irreducibility of boolean polynomials, which has been discussed in previous work by K. H. Kim and F. W. Roush (2005) and Y. Shitov (2014). We prove results about the irreducibility of polynomials and power series over the max-min semiring, a natural generalization of the boolean polynomials. We use combinatorial and probabilistic methods to prove that almost all polynomials are irreducible over the max-min semiring, generalizing work of Y. Shitov (2014) and proving a 2011 conjecture by D. L. Applegate, M. Le Brun, and N. J. A. Sloane. Furthermore, we use measure-theoretic methods and apply Borel's result on normal numbers to prove that almost all power series are asymptotically irreducible over the max-min semiring. This result generalizes work of E. Wirsing (1953).

math.CO↗

arXiv2021

The Real Schwarz Lemma: The Sequel

A decade ago, when teaching complex analysis, the third named author posed the question on whether or not there is an analogue to the Schwarz lemma for real analytic functions. This led to the note [MT], indicating that it is possible to have a real analytic automorphism $f$ of $(-1,1)$ with $f'(0)$ arbitrarily large. In this note we provide other families with this property, and moreover show that we can always find such a function so that $f'(0)$ equals any desired real number. We end with some questions on related problems.

math.CV↗

arXiv2021

The Generalized Bergman Game

Every positive integer may be written uniquely as a base-$β$ decomposition--that is a legal sum of powers of $β$--where $β$ is the dominating root of a non-increasing positive linear recurrence sequence. Guided by earlier work on a two-player game which produces the Zeckendorf Decomposition of an integer (see [Bai+19]), we define a broad class of two-player games played on an infinite tuple of non-negative integers which decompose a positive integer into its base-$β$ expansion. We call this game the Generalized Bergman Game. We prove that the longest possible Generalized Bergman game on an initial state $S$ with $n$ summands terminates in $Θ(n^2)$ time, and we also prove that the shortest possible Generalized Bergman game on an initial state terminates between $Ω(n)$ and $O(n^2)$ time. We also show a linear bound on the maximum length of the tuple used throughout the game.

math.NT↗