arXiv subjects

Constantin N. Beli

Publications and source records attributed to Constantin N. Beli.

4 recordsLinked to original sources

arXiv2022

Universal integral quadratic forms over dyadic local fields

A quadratic form over a non-archimedian local field of characteristic zero $F$ is called universal if it is integral and it represents all non-zero integers of $F$. Xu Fei and Zhang Yang determined all universal quadratic forms in the case when $F$ is non-dyadic. In the more complicated dyadic case, when $F$ is a finite extension of $\mathbb Q_2$, they solved the same problem only in the ternary case. In our paper we solve this problem in the general case. Our result is given in terms of BONGs (bases of norm generators) but in section 3 of the paper we translate (without a proof) our result in terms of the more traditional Jordan splittings. In the last section we give some results on $n$-universality. We show that it can be reduced to the cases $n\leq 4$ and we give explicit necessary conditions for $n$-universality in the case when $n\geq 3$, $n$ odd. (A quadratic form is called $n$-maximal if it is integral and it represents all non-degenerate integral quadratic forms of rank $n$.)

math.NT↗

arXiv2020

Four conjectures by Zhi-Hong Sun

We prove some results conjectured by Zhi-Hong Sun regarding the value $\mod p$ of $\varepsilon_d^{\frac{p-1}4}$, where $\varepsilon_d$ is a unit of norm $-1$ in some fields $\mathbb Q(\sqrt d)$, with $\left(\frac{-1}p\right) =\left(\frac dp\right) =1$. The answer is given in terms of how $p$ writes as $p=f(x,y)=u^2+v^2$, with $x,y,u,v\in\mathbb Z$, where $f$ is a certain quadratic form of determinant $-4d$.

math.NT↗

arXiv2017

On a Waring's problem for integral quadratic and hermitian forms

For each positive integer $n$, let $g_{\mathbb Z}(n)$ be the smallest integer such that if an integral quadratic form in $n$ variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of $g_{\mathbb Z}(n)$ squares of integral linear forms. We show that as $n$ goes to infinity, the growth of $g_{\mathbb Z}(n)$ is at most an exponential of $\sqrt{n}$. Our result improves the best known upper bound on $g_{\mathbb Z}(n)$ which is in the order of an exponential of $n$. We also define an analogous number $g_{\mathcal O}^*(n)$ for writing hermitian forms over the ring of integers $\mathcal O$ of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1, we show that the growth of $g_{\mathcal O}^*(n)$ is at most an exponential of $\sqrt{n}$. We also improve results of Conway-Sloane and Kim-Oh on $s$-integral lattices.

math.NT↗

arXiv2015

Dispersion for 1-d Scrodinger and wave equation with BV coefficients

In this paper we analyze the dispersion for one dimensional wave and Schrodinger equations with BV coefficients. In the case of the wave equation we give a complete answer in terms of the variation of the logarithm of the coefficient showing that dispersion occurs if this variation is small enough but it may fail when the variation goes beyond a sharp threshold. For the Schrodigner equation we prove that the dispersion holds under the same smallness assumption on the variation of the coefficient. But, whether dispersion may fail for larger coefficients is unknown for the Schrodinger equation.

math.AP↗