Related papers: Zeros of Systems of ${\mathfrak p}$-adic Quadratic…

This survey describes work on the number of variables required to ensure that a system of r quadratic forms over the p-adics has a non-trivial common zero.

It is shown that a quintic form over a p-adic field with at least 26 variables has a non-trivial zero, providing that the cardinality of the residue class field exceeds 9.

We verify a conjecture of Emil Artin, for the case of a Cubic and Quadratic form over any $p$-adic field, provided the cardinality of the residue class field exceeds 293. That is any Cubic and Quadratic form with at least 14 variables has a…

We show that all $p$-adic quintic forms in at least $n>4562911$ variables have a non-trivial zero. We also derive new result concerning systems of cubic and quadratic forms.

We show that a quartic $p$-adic form with at least $3192$ variables possesses a non-trivial zero. We also prove new results on systems of cubic, quadratic and linear forms. As an example, we show that for a system comprising two cubic forms…

A variant of Brauer's induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new…

We derive an upper bound for the least number of variables needed to guarantee that a system of t quadratic forms (t>=2) over a field F has a nontrivial zero. In particular, if F is a local field, then 2t^2+3 variables insure the existence…

Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than…

We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is…

We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite…

Let $K$ be a number field and $\mathfrak{p} \mid (2)$ be a prime ideal. We compute the fourth level of the $\mathfrak{p}$-adic completions of $K$ when the ramification index is $4$ and the inertial degree is trivial for the ideal…

In this paper, we study additively indecomposable quadratic forms over real biquadratic and simplest cubic fields. In particular, we show that over these fields, we can always find such a classical form in 2 variables, which differs from…

In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class…

In this paper, we will interest in finding the number of zeros of the quadratic forms over finite fields. We will apply the tool for finding the number of rational points of supersingular curves in [6]. We will give some more tools for…

We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion…

We provide a sufficient condition for solvability of a system of real quadratic equations $p_i(x)=y_i$, $i=1, \ldots, m$, where $p_i: {\mathbb R}^n \longrightarrow {\mathbb R}$ are quadratic forms. By solving a positive semidefinite…

A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary…

We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version…

It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders…

This is an exposition of work on Artin's Conjecture on the zeros of $p$-adic forms. A variety of lines of attack are described, going back to 1945. However there is particular emphasis on recent developments concerning quartic forms on the…