16,606 research outputs found
Equations in the Hadamard ring of rational functions
Let k be a number field. It is well known that the set of sequences composed
by Taylor coefficients of rational functions over k is closed under
component-wise operations, and so it can be equipped with a ring structure. A
conjecture due to Pisot asks if (after enlarging the field) one can take d-th
roots in this ring, provided d-th roots of coefficients can be taken in k. This
was proved true in a preceding paper of the second author; in this article we
generalize this result to more general equations, monic in Y, where the former
case can be recovered for g(X,Y)=X^d-Y=0. Combining this with the Hadamard
quotient theorem by Pourchet and Van der Poorten, we are able to get rid of the
monic restriction, and have a theorem that generalizes both results.Comment: 18 pages, LaTe
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
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