Related papers: Improvements in Birch's theorem on forms in many v…

We prove that an additive form of degree $d=2m$, $m$ odd, $m\ge3$, over the unramified quadratic extension $\mathbb{Q}_2(\sqrt{5})$ has a nontrivial zero if the number of variables $s$ satisifies $s \ge 4d+1$. If $3 \nmid d$, then there…

Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 2$, and $V_F^*$ the singular locus of the hypersurface $\{\mathbf{x} \in \mathbb{A}^n_{\mathbb{C}}: F(\mathbf{x}) = 0 \}$. A longstanding result of Birch…

Let $f_1, ..., f_R$ be rational forms of degree $d \ge 2$ in $n > \sigma + R(R+1)(d-1)2^{d-1}$ variables, where $\sigma$ is the dimension of the affine variety cut out by the condition $\mathrm{rank}(\nabla f_k)_{k=1}^R < R$. Assume that…

We prove that an additive form of degree $d=2m$, $m$ odd over any totally ramified extension of $\mathbb{Q}_2$ has a nontrivial zero if the number of variables $s$ satisifies $s \ge \frac{d^2}{4} + 3d + 1$.

Let $K$ be a Birch field, that is, a field for which every diagonal form of odd degree in sufficiently many variables admits a non-zero solution; for example, $K$ could be the field of rational numbers. Let $f_1, \ldots, f_r$ be homogeneous…

We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin-Peyre conjecture for a…

By adopting a new approach to the analysis of the density of p-adic solutions arising in applications of the circle method, we show that under modest conditions the existence of non-trivial p-adic solutions suffices to establish positivity…

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…

We show that the Fr\"oberg conjecture holds in the second non-trivial degree for an ideal generated by generic forms of degree $d>2$. We also show that the conjecture is true up to degree $2d-1$ provided that the number of variables is…

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 paper, we introduce a new and direct approach to study the solvability of systems of equations generated by bilinear forms. More precisely, let $B (\cdot, \cdot)$ be a non-degenerate bilinear form and $E$ be a set in…

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds…

We determine the minimal number of variables $\Gamma^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=2m$, $m$ odd, $m \ge 3$ over the six ramified quadratic extensions of $\mathbb{Q}_2$. We prove that if…

Let $p$ be an odd prime and $d = p^{\tau}(p-1)$. In the spirit of Aritn's conjecture, consider the system of two diagonal forms of degree $d$ in $s$ variables given by \begin{equation*}\begin{split} a_1x_1^d + \cdots + a_sx_s^d = 0\\…

A fundamental result of Springer says that a quadratic form over a field of characteristic not 2 is isotropic if it is so after an odd degree extension. In this paper we generalize Springer's theorem as follows. Let R be a an arbitrary…

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…

Let $B$ be a bilinear form on pairs of points in the complex plane, of the form $B(p,q) = p^TMq$, for an invertible $2\times2$ complex matrix $M$. We prove that any finite set $S$ contained in an irreducible algebraic curve $C$ of degree…

This document provides a formal proof of Birkhoff's completeness theorem for multi-sorted algebras which states that any equational entailment valid in all models is also provable in the equational theory. More precisely, if a certain…

Let $p$ be a prime and $b(x)$ be an irreducible polynomial of degree $k$ over $\mathbb{F}_p$. Let $d\geq 1$ be an integer. Consider the following question: Is $b(x^d)$ irreducible? We derive necessary conditions for $b(x^d)$ to be…

Let $P$ be a principal indecomposable module of a finite group $G$ in characteristic $2$ and let $\varphi$ be the Brauer character of the corresponding simple $G$-module. We show that $P$ affords a non-degenerate $G$-invariant quadratic…