Optimal density for values of generic polynomial maps
Optimal density for values of generic polynomial maps
复制标题
通用多项式映射值的最佳密度
DOI：
10.1353/ajm.2020.0049
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发表时间：
2018
影响因子：
1.7
通讯作者：
A. Nevo
中科院分区：
文献类型：
--
作者：
Anish Ghosh;A. Gorodnik;A. Nevo
摘要：我们建立了奥本海姆丢番图逼近问题最小积分解大小的最佳界$|Q（x）-\\xi| <\\n $对于一般的三进制形式$Q$是$|X|\\n\\n\\{-1\}$。我们还建立了许多其他自然问题中多项式映射值的最佳密度率，包括限制在适当二次曲面上的线性形式的值，以及由$M_3上的共轭不变多项式环的生成元定义的多项式映射的值（\\Bbb\{C\}）$。这些结果是我们开发的一般方法的实例，该方法考虑欧氏空间的有理仿射代数子簇，在半单李群G$的作用下是不变的和齐次的。给定定义在欧氏空间上的多项式映射F，它在作用群G的半单子群H下不变，考虑它的平移F\\circ g的族。我们研究了这些多项式函数的限制的整数点的品种局限于一个大的欧氏球。我们的主要结果建立了一个明确的密度率为他们的价值观，一般多项式的家庭。这个问题已经被广泛研究之前，当多项式的问题是线性的，在经典丢番图近似的上下文中，但很少有人知道它的多项式的更高的次数。我们制定了一个启发式的鸽子洞的密度下限和一个明确的上限，制定了一个充分条件的上下界的重合，并在一些自然的例子中建立，他们确实匹配。最后，我们还建立了齐次射影簇上的齐次多项式的值的密度率。
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.