For every quadratic number field $K$, we prove a uniform $3$-unknown Diophantine definition of integer tuples in $\mathcal{O}_K$, allowing finitely many polynomial nonvanishing conditions. This yields an effective $+3$ transfer principle and a $13$-unknown representation of every recursively enumerable integer relation. Consequently, there exists an absolute degree bound $D_0\geq1$ such that for every quadratic number field $K$, there is no algorithm that, given \[ P(Y_1,\ldots,Y_{13})\in\mathbb{Z}[Y_1,\ldots,Y_{13}],\quad \mathrm{deg}\ P\leq D_0, \] decides whether $P=0$ has a solution in $\mathcal{O}_K^{13}$. The arithmetic input is a fourth-order Pell--Lucas congruence. It is a specialization of the norm-one Lucas multiplication formula, which yields exact valuations for the deviation of a Lucas quotient from its linear term, together with deviation criteria for Lucas--Wieferich and Wall--Sun--Sun primes. We also establish local surjectivity and $\ell$-adic density for second-order correction terms for norm-one Lucas sequences.
Jungin Lee in 2018 proved a necessary and sufficient condition that an integral quadratic form $\sum_{i=1}^{m} a_iX_i^2$ is universal over $M_2(\mathbb{Z})$. For a positive integer $n \geq 2$, Lee defined $f(n)$ to be the smallest positive integer $m$ such that for every pairwise coprime $a_1, a_2, \ldots a_m \in \math...
We study triples of distinct nonconstant polynomials $a,b,c$ over $\mathbb{R}$, $\mathbb{Q}$ or $\mathbb{Z}$ for which $$ ab+1,\qquad ac+1,\qquad bc+1,\qquad abc+1 $$ are all polynomial squares. We show that, up to permutation and a common invertible affine change of variable, there is exactly one such triple of quadra...
Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $A,B,C \in \mathcal{O}_K \setminus\{0\}$. Denote by $S_K'$ the set of prime ideals of $\mathcal{O}_K$ dividing $2ABC$. Assuming two standard conjectures concerning the modularity of mod-$p$ Galois representations and the Eichler-Shimura correspond...
We prove an incidence bound for bipartite graphs on finite subsets of $\mathbb{F}^2\times \mathbb{F}^2$ defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is $K_{k,k}$-free and its vertex classes have sizes $m$ and $n$, then it has $O_{t,k}((mn)^{2/3}+m+n+mn/p)$ edges, where $t$...
We prove a quantitative polynomial Roth theorem for corners in \(\mathbb F_p^2\) for arbitrary pairs of linearly independent polynomials. More precisely, given a positive integer $d$, there are constants $p_0$ and $C$ (depending only on $p$) so that for every $ p>p_0$, if polynomials \(\phi_1,\phi_2\in \mathbb \mathbb{...
Ji Li, Chun-Yen Shen, T. Truong et al.· 0 citations
For a full-dimensional rational polytope $P\subset\mathbb{R}^d$ and a real dilation parameter $t>0$, the integer point enumerator is defined by $L_{P}(t):= |tP\cap\mathbb{Z}^d|$. We determine exactly which translation vectors $\mathbf y=(y_1,\ldots,y_d)\in\mathbb{R}^d$ have the property that the single translated count...
S. Robins· 0 citations
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