4,501 research outputs found
Arithmetic progressions in binary quadratic forms and norm forms
We prove an upper bound for the length of an arithmetic progression
represented by an irreducible integral binary quadratic form or a norm form,
which depends only on the form and the progression's common difference. For
quadratic forms, this improves significantly upon an earlier result of Dey and
Thangadurai.Comment: 7 pages; minor revision; to appear in BLM
A linear time algorithm for the orbit problem over cyclic groups
The orbit problem is at the heart of symmetry reduction methods for model
checking concurrent systems. It asks whether two given configurations in a
concurrent system (represented as finite strings over some finite alphabet) are
in the same orbit with respect to a given finite permutation group (represented
by their generators) acting on this set of configurations by permuting indices.
It is known that the problem is in general as hard as the graph isomorphism
problem, whose precise complexity (whether it is solvable in polynomial-time)
is a long-standing open problem. In this paper, we consider the restriction of
the orbit problem when the permutation group is cyclic (i.e. generated by a
single permutation), an important restriction of the problem. It is known that
this subproblem is solvable in polynomial-time. Our main result is a
linear-time algorithm for this subproblem.Comment: Accepted in Acta Informatica in Nov 201
Linear correlations amongst numbers represented by positive definite binary quadratic forms
Given a positive definite binary quadratic form f, let r(n) = |{(x,y):
f(x,y)=n}| denote its representation function. In this paper we study linear
correlations of these functions. For example, if r_1, ..., r_k are
representation functions, we obtain an asymptotic for sum_{n,d} r_1(n) r_2(n+d)
... r_k(n+ (k-1)d).Comment: 60 pages. Small correction
Binary linear forms over finite sets of integers
Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with
integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this
paper it is proved that for every pair of normalized binary linear forms
f(x,y)=u_1x+v_1y and g(x,y)=u_2x+v_2y with integral coefficients, there exist
arbitrarily large finite sets of integers A and B such that |f(A)| > |g(A)| and
|f(B)| < |g(B)|.Comment: 20 page
- …