Sums and differences of four k-th powers

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent results by Salberger on the density of integral points on affine surfaces. More generally we consider representations by any integral diagonal form. The upper bound has the form $O_{N}(B^{c/\sqrt{k}})$, whereas earlier versions of the Determinant method would produce an exponent for $B$ of order $k^{-1/3}$ in this case. Furthermore, we prove that the number of representations of a positive integer $N$ as a sum of four $k$-th powers of non-negative integers is at most $O_{\epsilon}(N^{1/k+2/k^{3/2}+\epsilon})$ for $k \geq 3$, improving upon bounds by Wisdom.Comment: 18 pages. Mistake corrected in the statement of Theorem 1.2. To appear in Monatsh. Mat

Violation of Cluster Decomposition and Absence of Light-Cones in Local Integer and Half-Integer Spin Chains

We compute the ground-state correlation functions of an exactly solvable chain of integer spins, recently introduced in [R. Movassagh and P. W. Shor, arXiv: 1408.1657], whose ground state can be expressed in terms of a uniform superposition of all colored Motzkin paths. Our analytical results show that for spin s >= 2 there is a violation of the cluster decomposition property. This has to be contrasted with s = 1, where the cluster property holds. Correspondingly, for s = 1 one gets a light-cone profile in the propagation of excitations after a local quench, while the cone is absent for s = 2, as shown by time dependent density-matrix renormalization group. Moreover, we introduce an original solvable model of half-integer spins, which we refer to as Fredkin spin chain, whose ground state can be expressed in terms of superposition of all Dyck paths. For this model we exactly calculate the magnetization and correlation functions, finding that for s = 1/2, a conelike propagation occurs, while for higher spins, s >= 3/2, the colors prevent any cone formation and clustering is violated, together with square root deviation from the area law for the entanglement entropy. \ua9 2016 American Physical Society

A generalization of the Bombieri-Pila determinant method

The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper we give a generalization of that method to varieties of higher dimension, yielding a proof of Heath-Brown's 'Theorem 14' by real-analytic considerations alone.Comment: 13 page

Singular del Pezzo surfaces that are equivariant compactifications

We determine which singular del Pezzo surfaces are equivariant compactifications of G_a^2, to assist with proofs of Manin's conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of a semidirect product of G_a and G_m.Comment: 14 pages, main result extended to non-closed ground field

Campana points of bounded height on vector group compactifications

We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version that delineates well different types of behaviour of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By importing work of Chambert-Loir and Tschinkel to our set-up, we prove a log version of Manin's conjecture for klt Campana points on equivariant compactifications of vector groups.Comment: 52 pages; minor revision, changes in the definition of Campana point

談本票裁定

For any N ≥ 2, let Z ⊂ ℙN be a geometrically integral algebraic variety of degree d. This article is concerned with the number Nz(B) of ℚ-rational points on Z which have height at most B. For any ε > 0, we establish the estimate NZ(B) = O d,ε,N(Bdim Z+ε), provided that d ≥ 6. As indicated, the implied constant depends at most on d, ε, and N