Related papers: Squares of positive $(p,p)$-forms

For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $\alpha$, there…

We provide a counter-example to Hutchinson's original proof of $C^{1,\alpha}$ representation of curvature $m$-varifolds with $L^q$-integrable second fundamental form and $q>m$ in [6]. We also provide an alternative proof of the same result…

We study sets of the form $A = \big\{ n \in \mathbb N \big| \lVert p(n) \rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$ for various real valued polynomials $p$ and decay rates $\varepsilon$. In particular, we ask when such sets…

We show that the Pythagoras number of rings of type $\mathbb{R}[x,y, \sqrt{f(x,y)}]$ is infinite, provided that the polynomial $f(x,y)$ satisfies some mild conditions.

It is a well known fact that, if $\Sigma$ is an Einstein hypersurface with positive scalar curvature, then it is a round sphere. We give a stable version of this result showing that if a hypersurface is almost Einstein in a $L^p$-sense,…

We provide a counterexample to some statements dealing with a sufficient property for the square of a matrix to be a $P_0^+$ -matrix.

We prove that for any prime number $p\ge 3$, there exists a positive number $\kappa_p$ such that $\chi(\mathcal{O}_X)\ge \kappa_pc_1^2$ holds true for all algebraic surfaces $X$ of general type in characteristic $p$. In particular,…

We prove that |{1<=y<=x: y is odd and not of the form p+2^a+2^b}|>>x^{1-\epsilon} for any \epsilon>0, where the implied constant only depends on \epsilon.

We prove Kitaoka's conjecture for all totally real number fields of degree 4 -- namely, there is no positive definite classical quadratic form in three variables which is universal. To achieve this, we study the fields (often without…

Let $f$ be a holomorphic modular form of prime level $p$ and trivial nebentypus. We show that there exists a computable $\delta>0$, such that $$ L\left(\tfrac{1}{2},\mathrm{Sym}^2 f\right)\ll p^{\tfrac{1}{2}-\delta}, $$ with the implied…

We construct examples of groups that are $FP_2(\mathbb{Q})$ and $FP_2(\mathbb{Z}/p\mathbb{Z})$ for all primes $p$ but not of type $FP_2(\mathbb{Z})$.

Let $f$ be a positive definite (non-classic) integral quaternary quadratic form. We say $f$ is strongly $s$-regular if it satisfies a regularity property on the number of representations of squares of integers. In this article, we prove…

In this paper we prove some results on the possible multiplicative orders of $\alpha + \alpha^{-1}$ when $\alpha$ is a non-zero element of a finite field of characteristic 2. The results of the paper rely on a previous investigation on the…

We consider parametric equations driven by the sum of a $p$-Laplacian and a Laplace operator (the so-called $(p,2)$-equations). We study the existence and multiplicity of solutions when the parameter $\lambda>0$ is near the principal…

In this paper, we examine the homotopy classes of positive loops in ${\rm Sp}(2n)$. We demonstrate that two positive loops are homotopic if and only if they are homotopic through positive loops. As consequences, we can extend several…

This is a note on the classical Waring's problem for several homogeneous forms. For positive integers (n,d,r,s), fix a general r-dimensional subspace of degree d forms in n+1 variables. We describe the family of s-sided polar polyhedra of…

For $0\leq \alpha<1$ and prime number $p$ let $L(\alpha,p)$ be the sum of the first $[\alpha p]$ values of Legendre symbol modulo $p$. We study positivity of $L(\alpha,p)$ and prove that for $|\alpha-\frac13|<2\cdot 10^{-6}$ and for…

Let $n$ be a positive integer. We study the diophantine equation $ab(ab-1)-na=\Delta^2$, where $a,b$ are positive integers. We also show that if a system of two congruences is soluble, then an equation which is a translation of…

A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the…

We find normal forms for del Pezzo surfaces of degree $2$ over algebraically closed fields of characteristic $2$. For each normal form, we describe the structure of the group of automorphisms of the surface. In particular, we classify all…