Aug 2026· 2 citations· ⚡ 1 influential· 13 references
Mathematics
Abstract
Let $K$ be any quadratic number field, and let $O_K$ be the ring of algebraic integers in $K$. In 1975 J. Denef proved that Hilbert's Tenth Problem over $O_K$ has a negative solution. In this paper we establish the following undecidability result: There is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{16})=0$ (with integer coefficients and 16 unknowns) has solutions over $O_K$. Moreover, when $K$ is a real quadratic field, we show that $15$ unknowns suffice for undecidability.
In this work, we show that if $h(x)$ is an irreducible monic integer polynomial of degree $2q$ (with $q$ prime), whose defining field extension of $\mathbb{Q}$ contains a Galois extension of degree $q$, then there is a positive density of integers $n$ such that $h(n)$ is squarefree; in particular, $h(n)$ is squarefree...
Let $K$ be an infinite field of characteristic $2$. It is well known that the Lie algebra $\mathfrak{gl}_2(K)=M_2(K)^{(-)}$ of all $2\times 2$ matrices does not admit a finite basis of polynomial identities. Similarly, the finite basis problem for the associative algebra $M_2(K)$ remains open. In this note, we obtain t...
We show that there are integers $N$ and odd $D$ such that no algorithm can decide, from the rational coefficients of a homogeneous polynomial vector field $F$ in dimension $N$ of degree $D$, whether the origin is Lyapunov stable for $\dot Y=F(Y)$. This proves a conjecture of V. I. Arnold. We provide a Lean formalizatio...
Let $K$ be a number field of degree $D$ with maximal order $\mathcal{O}_K$. We show that under certain conditions on $K$, which in particular are always satisfied if $D$ is odd or if $D \geq 3$ and $K$ is primitive, the set of positive integers $N_K$ that can be expressed as a sum of two units in $\mathcal{O}_K^*$ is a...
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...
Let $X$ be an algebraic variety of dimension $d$ over an algebraically closed field $k$ of zero characteristic. Suppose that $G$ is a virtually polycyclic subgroup in the group $\mathrm{Bir}(X)$ of birational automorphisms of $X$. We show that the virtual derived length of $G$ does not exceed $2d+1$. Moreover, if $X$ i...
A. Golota· 0 citations
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