Related papers: On binary cubic and quartic forms

We investigate monodromy groups arising in enumerative geometry, with a particular focus on how these groups are influenced by prescribed symmetries. To study these phenomena effectively, we work in the framework of moduli stacks rather…

For every positive integer k, it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in k arithmetic progressions. For k=1,…

We prove a spectral summation formula for the product of four Fourier coefficients of half-integral weight cusp forms in Kohnen's subspace. The other side of the formula involves certain generalized class numbers of pairs of quadratic forms…

We give an overview of universal quadratic forms and lattices, focusing on the recent developments over the rings of integers in totally real number fields. In particular, we discuss indecomposable algebraic integers as one of the main…

The aim of this note is to give a formula expressing the trace form associated with the 27 lines of a cubic surface.

We show that the binary representation of the integers has a role to play in many aspects of Clifford algebras.

We classify all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of cubic forms. We deduce a classification of any Galois cubic extension of a…

We show that the variety of symmetric implication algebras is generated from cubic implication algebras and Boolean algebras. We do this by developing the notion of a locally symmetric implication algebra that has properties similar to…

Bielliptic and quasi-bielliptic surfaces form one of the four classes of minimal smooth projective surfaces of Kodaira dimension $0$. In this article, we determine the automorphism schemes of these surfaces over algebraically closed fields…

Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y…

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer $N$, first we give the classification of positive…

The Russell cubic is a smooth contractible affine complex threefold which is not isomorphic to affine three-space. In previous articles, we discussed the structure of the automorphism group of this variety. Here we review some consequences…

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…

In this article, we develop new methods for counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations. We illustrate these methods for a representation of cardinal…

We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over the rationals. The estimate is uniform in the coefficients of the underlying quadratic form.

We show that any rational cubic hypersurface of dimension at least 33 defined over a number field $K$ vanishes on a $K$-rational projective line, reducing the previous lower bound of Wooley by two. For $K=\mathbb Q$ we can reduce the bound…

We study equivariant birationality from the perspective of derived categories. We produce examples of nonlinearizable but stably linearizable actions of finite groups on smooth cubic fourfolds.

We study the existence of cubic quotients of finite-dimensional quasi-normed spaces, that is, quotients well isomorphic to $\ell_{\infty}^k$ for some $k.$ We give two results of this nature. The first guarantees a proportional dimensional…

We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we…

A Lie-algebraic classification of the variable coefficient cubic-quintic nonlinear Schr\"odinger equations involving 5 arbitrary functions of space and time is performed under the action of equivalence transformations. It is shown that…