Related papers: Distinct values of bilinear forms on algebraic cur…

We prove a boundedness criterion for a class of dyadic multilinear forms acting on two-dimensional functions. Their structure is more general than the one of classical multilinear Calder\'{o}n-Zygmund operators as several functions can now…

A pair of symmetric bilinear forms A and B determine a binary form $f(x,y) = disc(Ax-By)$. We prove that the question of whether a given binary form can be written in this way as a discriminant form generically satisfies a local-global…

Let X be an algebraic curve, defined over a perfect field, and G a divisor on X. If X has sufficiently many points, we show how to construct a divisor D on X such that l(2D-G)=0, of essentially any degree such that this is compatible the…

In this paper, we prove that a binary definite quadratic form over F_q[t], where q is odd, is completely determined up to equivalence by the polynomials it represents up to degree 3m-2, where m is the degree of its discriminant. We also…

Given an integral variation of Hodge structure $\mathbb{V}$ on a complex algebraic variety $S$, polarized by some bilinear form $Q : \mathbb{V} \otimes \mathbb{V} \to \mathbb{Z}$, it is believed that the set $\mathcal{A}^{\textrm{iso}}_{0}…

We consider the question of determining whether two binary cubic forms over an arbitrary field $K$ whose characteristic is not $2$ or $3$ are equivalent under the actions of either GL$(2,K)$ or SL$(2,K)$, deriving two necessary and…

We prove that the set of CM points on the Shimura curve associated to an Eichler order inside an indefinite quaternion $\mathbb{Q}$-algebra, is in bijection with the set of certain classes of $p$-adic binary quadratic forms, where $p$ is a…

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by R\'edei-type polynomials, are powerful in studying blocking sets. In this paper, we…

The set of linear, differential operators preserving the vector space of couples of polynomials of degrees n and n-2 in one real variable leads to an abstract associative graded algebra A(2). The irreducible, finite dimensional…

The multiplicity of an algebraic curve $C$ in the complex plane at a point $p$ on that curve is defined as the number of points that occur at the intersection of $C$ with a general complex line that passes close to the point $p$. It is…

We prove a variant of the Sylvester-Gallai theorem for cubics (algebraic curves of degree three): If a finite set of sufficiently many points in $\mathbb{R}^2$ is not contained in a cubic, then there is a cubic that contains exactly nine of…

We give new improvements to the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in extensions of finite fields through interpolation on algebraic curves. Our approach features three…

Let $K$ be a finite abelian group and let $\exp(K)$ denote the least common multiple of the orders of the elements of $K$. A $BH(K,h)$ matrix is a $K$-invariant $|K|\times |K|$ matrix $H$ whose entries are complex $h$th roots of unity such…

In this paper we compute the number of n degree representations of a group of order p^3 for p an odd prime and the dimensions of corresponding spaces of invariant bilinear forms over an algebraically closed field F. We explicitly discuss…

For every absolutely irreducible orthogonal representation of a twisted form of SL2 over a field of characteristic zero, we compute the "unique" symmetric bilinear form that is invariant under the group action. We also prove the analogous…

Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C:y^2=f(x)$ the corresponding genus $g$…

We show, assuming Schanuel's conjecture, that every irreducible complex polynomial in two variables where both variables appear has infinitely many algebraically independent solutions of the form (z,e^z).

We show how the proof of the Grothendieck Theorem for jointly completely bounded bilinear forms on C*-algebras by Haagerup and Musat can be modified in such a way that the method of proof is essentially C*-algebraic. To this purpose, we use…

Given a quaternionic form G of a p-adic classical group (p odd) we classify all cuspidal irreducible representations of G with coefficients in an algebraically closed field of characteristic different from p. We prove two theorems: At…

Let $G$ be a connected semisimple simply connected Lie group with a compact Cartan subgroup and let $\Gamma$ be a uniform lattice in $G$. Let $\widehat{G}_d$ denote the set of equivalence classes of unitary discrete series representations…