More Like this

1. Super-polynomial accuracy of one dimensional randomized nets using the median of means

   https://doi.org/10.1090/mcom/3791  

   Pan, Zexin; Owen, Art (March 2023, Mathematics of Computation)

   Let $f$be analytic on $[0,1]$with $|f^{\left(k\right)}\left(1/2\right)|⩽<\#comment/>A{\mathrm{\alpha <\#comment/>}}^{k}k!$for some constants $A$and $\mathrm{\alpha <\#comment/>}>2$and all $k⩾<\#comment/>1$. We show that the median estimate of $\mathrm{\mu <\#comment/>}={\int <\#comment/>}_0^{1}f\left(x\right)\mathrm{d}x$under random linear scrambling with $n=2^m$points converges at the rate $O\left(n^{-<\#comment/>c\log <\#comment/>\left(n\right)}\right)$for any $c>3\log <\#comment/>\left(2\right)/{\mathrm{\pi <\#comment/>}}^2\approx <\#comment/>0.21$. We also get a super-polynomial convergence rate for the sample median of $2k-<\#comment/>1$random linearly scrambled estimates, when $k/m$is bounded away from zero. When $f$has a $p$’th derivative that satisfies a $\mathrm{\lambda <\#comment/>}$-Hölder condition then the median of means has error $O\left(n^{-<\#comment/>(p+\mathrm{\lambda <\#comment/>})+\mathrm{ϵ<\#comment/>}}\right)$for any $\mathrm{ϵ<\#comment/>}>0$, if $k\to <\#comment/>\mathrm{\infty <\#comment/>}$as $m\to <\#comment/>\mathrm{\infty <\#comment/>}$. The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number. 

   more » « less
2. Uniform Lech’s inequality

   https://doi.org/10.1090/proc/16304  

   Ma, Linquan; Smirnov, Ilya (February 2023, Proceedings of the American Mathematical Society)

   Let $(R,\mathfrak{m})$be a Noetherian local ring of dimension $d\ge <\#comment/>2$. We prove that if $e\left({\stackrel{^<\#comment/>}{R}}_{red}\right)>1$, then the classical Lech’s inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists $\mathrm{\epsilon <\#comment/>}>0$such that $e\left(I\right)\le <\#comment/>d!\left(e\right(R)-<\#comment/>\mathrm{\epsilon <\#comment/>})\mathrm{\ell <\#comment/>}\left(R/I\right)$for all $\mathfrak{m}$-primary ideals $I\subseteq <\#comment/>R$. This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of $I$. 

   more » « less
3. Regularity, singularities and ℎ-vector of graded algebras

   https://doi.org/10.1090/tran/9089  

   Dao, Hailong; Ma, Linquan; Varbaro, Matteo (January 2024, Transactions of the American Mathematical Society)

   Let $R$be a standard graded algebra over a field. We investigate how the singularities of $\mathrm{Spec}<\#comment/>R$or $\mathrm{Proj}<\#comment/>R$affect the $h$-vector of $R$, which is the coefficient of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if $R$satisfies Serre’s condition $\left(S_r\right)$and has reasonable singularities (Du Bois on the punctured spectrum or $F$-pure), then $h_0$, …, $h_r\ge <\#comment/>0$. Furthermore the multiplicity of $R$is at least $h_0+h_1+\cdots <\#comment/>+h_{r-<\#comment/>1}$. We also prove that equality in many cases forces $R$to be Cohen-Macaulay. The main technical tools are sharp bounds on regularity of certain $\mathrm{Ext}$modules, which can be viewed as Kodaira-type vanishing statements for Du Bois and $F$-pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be Cohen-Macaulay. Our results build on and extend previous work by de Fernex-Ein, Eisenbud-Goto, Huneke-Smith, Murai-Terai and others. 

   more » « less
4. A modular construction of unramified 𝑝-extensions of ℚ(ℕ^{1/𝕡})

   https://doi.org/10.1090/bproc/141  

   Lang, Jaclyn; Wake, Preston (October 2022, Proceedings of the American Mathematical Society, Series B)

   We show that for primes $N,p\ge <\#comment/>5$with $N\equiv <\#comment/>-<\#comment/>1modp$, the class number of $\mathbb{Q}\left(N^{1/p}\right)$is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N\equiv <\#comment/>-<\#comment/>1modp$, there is always a cusp form of weight $2$and level ${\mathrm{\Gamma <\#comment/>}}_0\left(N^2\right)$whose $\mathrm{\ell <\#comment/>}$th Fourier coefficient is congruent to $\mathrm{\ell <\#comment/>}+1$modulo a prime above $p$, for all primes $\mathrm{\ell <\#comment/>}$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree- $p$extension of $\mathbb{Q}\left(N^{1/p}\right)$. 

   more » « less
5. Compatible ideals in ℚ-Gorenstein rings

   https://doi.org/10.1090/proc/16331  

   Polstra, Thomas; Schwede, Karl (October 2023, Proceedings of the American Mathematical Society)

   Suppose $R$is a $F$-finite and $F$-pure $\mathbb{Q}$-Gorenstein local ring of prime characteristic $p>0$. We show that an ideal $I\subseteq <\#comment/>R$is uniformly compatible ideal (with all $p^{-<\#comment/>e}$-linear maps) if and only if exists a module finite ring map $R\to <\#comment/>S$such that the ideal $I$is the sum of images of all $R$-linear maps $S\to <\#comment/>R$. In other words, the set of uniformly compatible ideals is exactly the set of trace ideals of finite ring maps. 

   more » « less