Abstract:

We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $|Q(x)-\\xi|<\\epsilon$ for a generic ternary form $Q$ is $|x|\\ll\\epsilon^\{-1\}$. We also establish an optimal rate of density for the values of polynomials maps in a number of other natural problems, including the values of linear forms restricted to suitable quadratic surfaces, and the values of the polynomial map defined by the generators of the ring of conjugation-invariant polynomials on $M_3(\\Bbb\{C\})$.

These results are instances of a general approach that we develop, which considers a rational affine algebraic subvariety of Euclidean space, invariant and homogeneous under an action of a semisimple Lie group $G$. Given a polynomial map $F$ defined on the Euclidean space which is invariant under a semisimple subgroup $H$ of the acting group $G$, consider the family of its translates $F\\circ g$ by elements of the group. We study the restriction of these polynomial functions to the integer points on the variety confined to a large Euclidean ball. Our main results establish an explicit rate of density for their values, for generic polynomials in the family. This problem has been extensively studied before when the polynomials in question are linear, in the context of classical Diophantine approximation, but very little was known about it for polynomial of higher degree. We formulate a heuristic pigeonhole lower bound for the density and an explicit upper bound for it, formulate a sufficient condition for the coincidence of the lower and upper bounds, and in a number of natural examples establish that they indeed match. Finally, we also establish a rate of density for values of homogeneous polynomials on homogeneous projective varieties.

摘要：

我们建立了一般三元形式$ Q $的Oppenheim Diophantine逼近问题$ | Q（x）-\\ xi | <\\ epsilon $的最小积分解的大小的最佳界是$ | x | \ \ ll \\ epsilon ^ \ {-1 \} $。我们还为许多其他自然问题中的多项式图的值（包括限制在合适的二次曲面上的线性形式的值）以及由共轭环的生成器定义的多项式图的值确定了最佳密度比率-$ M_3（\\ Bbb \ {C \}）$上的不变多项式。

这些结果是我们开发的一种通用方法的实例，该方法考虑了一个半简单李群$ G $的作用下欧几里德空间的有理仿射代数子变量，不变和同质。给定在欧几里得空间上定义的多项式映射$ F $在表演组$ G $的半简单子组$ H $下是不变的，请考虑按其元素将其平移$ F \\ circ g $的族。我们研究了将这些多项式函数限制在局限于一个大型欧几里得球上的各种整数点上的情况。我们的主要结果为族中的一般多项式确定了其值的显式密度速率。在经典多项式Diophantine逼近的情况下，在所讨论的多项式为线性之前，已经对该问题进行了广泛的研究，但是对高阶多项式知之甚少。我们为密度制定了启发式信鸽下限，并为其明确设定了上限，为上下限的重合制定了充分条件，并在许多自然实例中确定它们确实匹配。最后，我们还建立了同质投影变种上同质多项式值的密度比率。