关于二次型所表示的多项式值
On the polynomial values represented by binary quadratic forms
- University of Leicester（莱斯特大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究对于给定非退化二次型\(F\)和整系数单变量多项式\(P\)，\(P(x)\)能否被\(F\)无限次表示的问题。开发方法并应用于特定多项式，得出\(x^6 - 4\)无限次是两平方和，解决相关丢番图方程整数解问题，还列出新的未解决最短方程。
AI中文摘要：
许多丢番图方程可归结为对于给定的非退化二次型\(F\)和整系数单变量多项式\(P\)，\(P(x)\)是否能对无穷多个\(x\)值由\(F\)表示的问题。我们针对某些三次和四次多项式\(P\)以及某些形如\(P(x)=R(Q(x))\)（其中\(R(t)\)和\(Q(x)\)分别是三次和二次多项式）开发了一种回答此问题的方法。通过将该方法应用于\(F(y,z)=y^2 + z^2\)，\(R(t)=t^3 - 4\)和\(Q(x)=x^2\)，得出\(x^6 - 4\)无限多次是两个平方数之和，进而意味着方程\(y^2 + x^3y + z^2 + 1 = 0\)有无限多个整数解。在此工作之前，这是关于其整数解集是有限还是无限尚属未知的最短方程。最后列出了有限性问题仍未解决的新的最短方程列表。本文所有主要结果已在Lean中使用亚里士多德逻辑形式化。
英文摘要：
Many Diophantine equations can be reduced to the question of whether, for a given non-degenerate integral binary quadratic form $F$ and a univariate polynomial $P$ with integer coefficients, $P(x)$ can be represented by $F$ for infinitely many values of $x$. We develop a method for answering this question for certain cubic and quartic polynomials $P$, as well as for certain polynomials of the form $P(x)=R(Q(x))$, where $R(t)$ and $Q(x)$ are polynomials of degrees $3$ and $2$, respectively. Applying this method with $F(y,z)=y^2+z^2$, $R(t)=t^3-4$ and $Q(x)=x^2$, we conclude that $x^6-4$ is a sum of two squares infinitely often. In turn, this implies that the equation $y^2+x^3y+z^2+1=0$ has infinitely many integer solutions. Prior to this work, it was the shortest equation for which it was unknown whether its integer solution set is finite or infinite. We conclude with a list of the new shortest equations for which the finiteness problem remains open. All main results of this paper have been formalized in Lean using Aristotle.
