Incomplete Quadratic Exponential Sums in Several Variables

We consider incomplete exponential sums in several variables of the form S(f,n,m) = \frac{1}{2^n} \sum_{x_1 \in \{-1,1\}} ... \sum_{x_n \in \{-1,1\}} x_1 ... x_n e^{2\pi i f(x)/p}, where m>1 is odd and f is a polynomial of degree d with coefficients in Z/mZ. We investigate the conjecture, originating in a problem in computational complexity, that for each fixed d and m the maximum norm of S(f,n,m) converges exponentially fast to 0 as n grows to infinity. The conjecture is known to hold in the case when m=3 and d=2, but existing methods for studying incomplete exponential sums appear to be insufficient to resolve the question for an arbitrary odd modulus m, even when d=2. In the present paper we develop three separate techniques for studying the problem in the case of quadratic f, each of which establishes a different special case of the conjecture. We show that a bound of the required sort holds for almost all quadratic polynomials, a stronger form of the conjecture holds for all quadratic polynomials with no more than 10 variables, and for arbitrarily many variables the conjecture is true for a class of quadratic polynomials having a special form.Comment: 31 pages (minor corrections from original draft, references to new results in the subject, publication information

On Tractable Exponential Sums

We consider the problem of evaluating certain exponential sums. These sums take the form $\sum_{x_1,...,x_n \in Z_N} e^{f(x_1,...,x_n) {2 \pi i / N}}$, where each x_i is summed over a ring Z_N, and f(x_1,...,x_n) is a multivariate polynomial with integer coefficients. We show that the sum can be evaluated in polynomial time in n and log N when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree \ge 3, we show the problem is #P-hard, even for any fixed prime or prime power modulus. This leads to a complexity dichotomy theorem - a complete classification of each problem to be either computable in polynomial time or #P-hard - for a class of exponential sums. These sums arise in the classifications of graph homomorphisms and some other counting CSP type problems, and these results lead to complexity dichotomy theorems. For the polynomial-time algorithm, Gauss sums form the basic building blocks. For the hardness results, we prove group-theoretic necessary conditions for tractability. These tests imply that the problem is #P-hard for even very restricted families of simple cubic polynomials over fixed modulus N

Average Bateman--Horn for Kummer polynomials

For any $r \in \mathbb{N}$ and almost all $k \in \mathbb{N}$ smaller than $x^r$, we show that the polynomial $f(n) = n^r + k$ takes the expected number of prime values as $n$ ranges from 1 to $x$. As a consequence, we deduce statements concerning variants of the Hasse principle and of the integral Hasse principle for certain open varieties defined by equations of the form $N_{K/\mathbb{Q}}(\mathbf{z}) = t^r +k \neq 0$ where $K/\mathbb{Q}$ is a quadratic extension. A key ingredient in our proof is a new large sieve inequality for Dirichlet characters of exact order $r$.Comment: V2: Minor correction

Entropy in Dimension One

This paper completely classifies which numbers arise as the topological entropy associated to postcritically finite self-maps of the unit interval. Specifically, a positive real number h is the topological entropy of a postcritically finite self-map of the unit interval if and only if exp(h) is an algebraic integer that is at least as large as the absolute value of any of the conjugates of exp(h); that is, if exp(h) is a weak Perron number. The postcritically finite map may be chosen to be a polynomial all of whose critical points are in the interval (0,1). This paper also proves that the weak Perron numbers are precisely the numbers that arise as exp(h), where h is the topological entropy associated to ergodic train track representatives of outer automorphisms of a free group.Comment: 38 pages, 15 figures. This paper was completed by the author before his death, and was uploaded by Dylan Thurston. A version including endnotes by John Milnor will appear in the proceedings of the Banff conference on Frontiers in Complex Dynamic

Symmetric coupling of angular momenta, quadratic algebras and discrete polynomials

Eigenvalues and eigenfunctions of the volume operator, associated with the symmetric coupling of three SU(2) angular momentum operators, can be analyzed on the basis of a discrete Schroedinger-like equation which provides a semiclassical Hamiltonian picture of the evolution of a `quantum of space', as shown by the authors in a recent paper. Emphasis is given here to the formalization in terms of a quadratic symmetry algebra and its automorphism group. This view is related to the Askey scheme, the hierarchical structure which includes all hypergeometric polynomials of one (discrete or continuous) variable. Key tool for this comparative analysis is the duality operation defined on the generators of the quadratic algebra and suitably extended to the various families of overlap functions (generalized recoupling coefficients). These families, recognized as lying at the top level of the Askey scheme, are classified and a few limiting cases are addressed.Comment: 10 pages, talk given at "Physics and Mathematics of Nonlinear Phenomena" (PMNP2013), to appear in J. Phys. Conf. Serie