Values at integers of homogeneous polynomials
Berry and Tabor (1977) conjectured that the local spacings between the eigenvalues of the quantization of the Hamiltonian for generic, completely integrable systems follow the distribution of random numbers. \textit{P. Sarnak} [CMS Conf. Proc. 21, 181-203 (1997; Zbl 0911.11032)] considered the specific case of geodesic flow on a flat two-dimensional torus for which the Berry-Tabor conjecture reduces to a conjecture about the spacings between the values at non-negative integers of binary quadratic forms. In this paper, the spacings of the values at non-negative integers of homogeneous forms of higher degree have been considered. The results proved here are: (i) Given a compact set \(B\) that is the support of a smooth function, let \(k> 2\), and consider the set of homogeneous polynomials of degree \(k\) in \(k\) variables \[ P_\alpha (\overline{x})= \alpha_1 x_1^k+ \cdots+ \alpha_k x_k^k+ \cdots+ \alpha_d x_1 x_2\dots x_k. \] There is a set \(M\) of full measure in \(\mathbb{R}^d\) so that if \((\alpha_1,\dots, \alpha_d)\in M\), \(R_{p_\alpha} (a,b,B,T)\) approaches \(C_{p_\alpha}^2 (b-a)\) for all \(a< b\) where \(R_{p_\alpha}\) is the pair correlation of \(P\) and \(C_p\) is the inverse of the average distance between consecutive values of \(P\). (ii) Let \(P= \alpha x^3+ \beta y^3+ \gamma z^3\) be a fixed diagonal cubic form with non-zero coefficients and let \(P_i= q_{i1} x^3+ q_{i2} y^3+ q_{i3} z^3\) be a sequence of integer forms with \(q_{i1}/ q_{i2}\) and \(q_{i1}/ q_{i3}\) approaching \(\alpha/\beta\) and \(\alpha/\gamma\), respectively, as \(q_{i1}\) goes to infinity, with \(\text{gcd} (q_{i1}, q_{i2}, q_{i3})= 1\). Then the Hardy-Littlewood predictions for \(R_p(a,b,B,T)\) converge to \(C_p^2 (b-a)\). (iii) Homogeneous \(k\)th degree forms whose pair correlations approach delta functions are dense among those forms that factor into \(k\) linear terms over the complex numbers. The following conjectures along with their numerical evidence have also been given in this paper. (a) The number of integer solutions to \(\alpha (x_0^3- x_1^3)+ \beta (y_0^3-y_1^3)+ \gamma (z_0^3- z_1^3)= k\) satisfies Hardy-Littlewood estimates if one does not count the points on solution subvarieties. (b) Let \(k\) be larger than 2, and \(B\) a compact set that is the support of a smooth function, desymmetrized if \(k\) is even. If \(P\) is a homogeneous non-singular \(k\)th degree form in \(k\) variables not proportional to rational forms then \(E_p(a,b,B,T)\) the consecutive spacing distribution and \(R_p (a,b,B,T)\) approach the limits expected for random numbers.
eigenvalues of quantization of Hamiltonian
scientific article; zbMATH DE number 1425200