arXiv subjects

Qimin Song

Publications and source records attributed to Qimin Song.

3 recordsLinked to original sources

arXiv2025

Finiteness of Pythagoras numbers of finitely generated real algebras

In this paper, we establish two finiteness results and propose a conjecture concerning the Pythagoras number $P(A)$ of a finitely generated real algebra $A$. Let $X \hookrightarrow \mathbb{P}^n$ be an integral projective surface over $\mathbb{R}$, let $\widetilde{X}$ be the normalization of $X$, and let $s \in \Gamma(X,\mathcal{O}_X(1))$ be a nonzero section such that $\bigl(\widetilde{X}_{s=0}\bigr)^{\mathrm{red}}$ is formally real. We prove $P\bigl(\Gamma(X_{s\neq 0})\bigr)=\infty$. As a corollary, the Pythagoras numbers of integral smooth affine curves over $\mathbb{R}$ are shown to be unbounded. For any finitely generated $\mathbb{R}$-algebra $A$, if the Zariski closure of the real points of $\mathrm{Spec}(A)$ has dimension less than two, we demonstrate $P(A)<\infty$.

math.NT↗

arXiv2025

On imaginary quadratic fields with non-cyclic class groups

For a fixed abelian group $H$, let $N_H(X)$ be the number of square-free positive integers $d\leq X$ such that H is a subgroup of $CL(\mathbb{Q}(\sqrt{-d}))$. We obtain asymptotic lower bounds for $N_H(X)$ as $X\to\infty$ in two cases: $H=\mathbb{Z}/g_1\mathbb{Z}\times (\mathbb{Z}/2\mathbb{Z})^l$ for $l\geq 2$ and $2\nmid g_1\geq 3$, $H=(\mathbb{Z}/g\mathbb{Z})^2$ for $2\nmid g\geq 5$. More precisely, for any $\epsilon >0$, we showed $N_H(X)\gg X^{\frac{1}{2}+\frac{3}{2g_1+2}-\epsilon}$ when $H=\mathbb{Z}/g_1\mathbb{Z}\times (\mathbb{Z}/2\mathbb{Z})^l$ for $l\geq 2$ and $2\nmid g_1\geq 3$. For the second case, under a well known conjecture for square-free density of integral multivariate polynomials, for any $\epsilon >0$, we showed $N_H(X)\gg X^{\frac{1}{g-1}-\epsilon}$ when $H=(\mathbb{Z}/g\mathbb{Z})^2$ for $ g\geq 5$. The first case is an adaptation of Soundararajan's results for $H=\mathbb{Z}/g\mathbb{Z}$, and the second conditionally improves the bound $X^{\frac{1}{g}-\epsilon}$ due to Byeon and the bound $X^{\frac{1}{g}}/(\log X)^{2}$ due to Kulkarni and Levin.

math.NT↗

arXiv2024

Divisibility of class numbers of quadratic fields and a conjecture of Iizuka

Assume $x,\ y,\ n$ are positive integers and $n$ is odd. In this note, we show that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{x^{2}-y^{n}})$ is divisible by $n$ for fixed $x, n$ if $\gcd(2x,y)=1$ and $y>C$ where $C$ is a constant depending only on $x$ and $n$. Based on this result, for any odd integer $n$ and any positive integer $m$, we construct an infinite family of $m+1$ successive imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1^{2}})$, $\cdots$, $\mathbb{Q}(\sqrt{d+m^{2}})$ $(d\in \mathbb{Z})$ whose class numbers are all divisible by $n$.

math.NT↗