Related papers: A polynomial bound for the arithmetic $k$-cycle re…

Let $K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $K$ whose $\ell$-power torsion fields are arithmetically constrained for some rational prime $\ell$. Such arithmetic…

For positive integers $n>k$, let $P_{n,k}(x)=\displaystyle\sum_{j=0}^k \binom{n}{j}x^j $ be the polynomial obtained by truncating the binomial expansion of $(1+x)^n$ at the $k^{th}$ stage. These polynomials arose in the investigation of…

Let $A$ be a nonempty subset of finite abelian group $G$ of order $n$. For an integer $h \geq 2$, the restricted $h$-fold sumset $h^\wedge A$ is the set of all sums of $h$ distinct elements of $A$. It is known that if $G$ is a group of…

We use geometric and cohomological methods to show that given a degree bound for membership in ideals of a fixed degree type in the polynomial ring P=k[x_0,..., x_d], one obtains a good generic degree bound for membership in the tight…

The Gowers U^3 norm is one of a sequence of norms used in the study of arithmetic progressions. If G is an abelian group and A is a subset of G then the U^3(G) of the characteristic function 1_A is useful in the study of progressions of…

We use intersection theory, degeneration techniques and jet schemes to study log canonical thresholds. Our first result gives a lower bound for the log canonical threshold of a pair in terms of the log canonical threshold of the image by a…

We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n…

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F_q come either from Serre's refinement of the Weil bound if the genus…

For every natural number k we prove a decomposition theorem for bounded measurable functions on compact abelian groups into a structured part, a quasi random part and a small error term. In this theorem quasi randomness is measured with the…

We prove Hypothesis H in full generality for ${\rm GL}_n$ over any number field. This result is a consequence of our stronger effective bound on Euler products involving Rankin--Selberg coefficients at prime ideal powers. The proof rests on…

We establish an optimal (topological) irreducibility criterion for $p$-adic Banach principal series of $\mathrm{GL}_{n}(F)$, where $F/\mathbb{Q}_p$ is finite and $n \le 3$. This is new for $n = 3$ as well as for $n = 2$, $F \ne…

Let $\alpha$ be an algebraic number of degree $d\ge 3$ and let $K$ be the algebraic number field $\Q(\alpha)$. When $\varepsilon$ is a unit of $K$ such that $\Q(\alpha\varepsilon)=K$, we consider the irreducible polynomial $f_\varepsilon(X)…

Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek lower bounds for $P(k)$. In particular,…

Given a global field $K$ and a positive integer $n$, we present a diophantine criterion for a polynomial in one variable of degree $n$ over $K$ not to have any root in $K$. This strengthens the known result that the set of non-$n$-th-powers…

For a relative effective divisor $\mathcal{C}$ on a smooth projective family of surfaces $q:\mathcal{S}\rightarrow B$, we consider the locus in $B$ over which the fibres of $\mathcal{C}$ are $\delta$-nodal curves. We prove a conjecture by…

We give several generalisations of the Deuring reduction criterion for elliptic curves to abelian varieties of higher dimension. In particular the Newton polygon of the reduction of an abelian variety A with complex multiplication by F at a…

We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus $g$. We prove that the growth is bounded by a polynomial function of degree $14+\varepsilon$ for any $\varepsilon>0$ and, for…

This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $\lambda\left( G\right) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $k\geq1,$…

We take a unifying and new approach toward polynomial and trigonometric approximation in an arbitrary number of variables, resulting in a precise and general ready-to-use tool that anyone can easily apply in new situations of interest. The…

Let $f_1,\cdots,f_r\in k[x_1,\cdots,x_n]$ be homogeneous polynomial of degree $d$. Ananyan and Hochster (2016) proved that there exists a bound $N=N(r,d)$ where if collective strength of $f_1,\cdots,f_r\geq N$, then $f_1,\cdots,f_r$ are…