10,525 research outputs found
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 ,
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 and almost all smaller than
, we show that the polynomial takes the expected number
of prime values as ranges from 1 to . 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
where is a
quadratic extension. A key ingredient in our proof is a new large sieve
inequality for Dirichlet characters of exact order .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
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