We consider an equation of the type $$a_1^{}{x}_1^2+\cdots +a_n^{}{x}_n^2=b{x}_1\cdots x_n$$ over the finite field ${𝔽}_q={𝔽}_{p^s}$. Carlitz obtained formulas for the number of solutions to this equation when $n=3$ and when $n=4$ and $q\equiv 3(mod4)$. In our earlier papers, we found formulas for the number of solutions when $d=gcd(n-2,(q-1)/2)=1$ or $2$ or $4$; and when $d\>1$ and $-1$ is a power of $p$ modulo $2d$. In this paper, we obtain formulas for the number of solutions when $d=2^t$, $t\ge 3$, $p\equiv 3\text{or}5(mod8)$ or $p\equiv 9(mod16)$. For general case, we derive lower bounds for the number of solutions.

Let $p\ge 3$ be a prime. Let $n\in \mathbb{N}$ such that $n\ge 1$, let ${\chi }_1,...,{\chi }_n$ be characters of conductor $d$ not divided by $p$ and let $\omega$ be the Teichmüller character. For all $i$ between $1$ and $n$, for all $j$ between $0$ and $(p-3)/2$, set $${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_i{\omega }^{2j+1}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{odd};\hfill \\ {\chi }_i{\omega }^{2j}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$ Let $K={\mathbb{Q}}_p({\chi }_1,...,{\chi }_n)$ and let $\pi$ be a prime of the valuation ring ${𝒪}_K$ of $K$. For all $i,j$ let $f(T,{\theta }_{i,j})$ be the Iwasawa series associated to ${\theta }_{i,j}$ and $\overline{f(T,{\theta }_{i,j})}$ its reduction modulo $\left(\pi \right)$. Finally let $\overline{{𝔽}_p}$ be an algebraic closure of ${𝔽}_p$. Our main result is that if the characters ${\chi }_i$ are all distinct modulo $\left(\pi \right)$, then $1$ and the series...

Let $A,B$ denote generic binary forms, and let ${𝔲}_r={(A,B)}_r$ denote their $r$-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the $\left\{{𝔲}_r\right\}$. As a consequence, we show that each of the higher transvectants $\{{𝔲}_r:r\ge 2\}$ is redundant in the sense that it can be completely recovered from ${𝔲}_0$ and ${𝔲}_1$. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of $S{L}_2$-representations,...

We give a proof of an integral formula of Berndtsson which is related to the inversion of Fourier-Laplace transforms of $\overline{\partial }$-closed $(n,n-1)$-forms in the complement of a compact convex set in ${ℂ}^n$.

We provide a generalization of Scholz’s reciprocity law using the subfields $K_{2^{t-1}}$ and $K_{2^t}$ of $\mathbb{Q}\left({\zeta }_p\right)$, of degrees $2^{t-1}$ and $2^t$ over $\mathbb{Q}$, respectively. The proof requires a particular choice of primitive element for $K_{2^t}$ over $K_{2^{t-1}}$ and is based upon the splitting of the cyclotomic polynomial ${\Phi }_p\left(x\right)$ over the subfields.