127 research outputs found
Factoring bivariate lacunary polynomials without heights
We present an algorithm which computes the multilinear factors of bivariate
lacunary polynomials. It is based on a new Gap Theorem which allows to test
whether a polynomial of the form P(X,X+1) is identically zero in time
polynomial in the number of terms of P(X,Y). The algorithm we obtain is more
elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on
the valuation of polynomials of the previous form instead of the height of the
coefficients. As a result, it can be used to find some linear factors of
bivariate lacunary polynomials over a field of large finite characteristic in
probabilistic polynomial time.Comment: 25 pages, 1 appendi
Detecting lacunary perfect powers and computing their roots
We consider solutions to the equation f = h^r for polynomials f and h and
integer r > 1. Given a polynomial f in the lacunary (also called sparse or
super-sparse) representation, we first show how to determine if f can be
written as h^r and, if so, to find such an r. This is a Monte Carlo randomized
algorithm whose cost is polynomial in the number of non-zero terms of f and in
log(deg f), i.e., polynomial in the size of the lacunary representation, and it
works over GF(q)[x] (for large characteristic) as well as Q[x]. We also give
two deterministic algorithms to compute the perfect root h given f and r. The
first is output-sensitive (based on the sparsity of h) and works only over
Q[x]. A sparsity-sensitive Newton iteration forms the basis for the second
approach to computing h, which is extremely efficient and works over both
GF(q)[x] (for large characteristic) and Q[x], but depends on a number-theoretic
conjecture. Work of Erdos, Schinzel, Zannier, and others suggests that both of
these algorithms are unconditionally polynomial-time in the lacunary size of
the input polynomial f. Finally, we demonstrate the efficiency of the
randomized detection algorithm and the latter perfect root computation
algorithm with an implementation in the C++ library NTL.Comment: to appear in Journal of Symbolic Computation (JSC), 201
Rigidity and Non-recurrence along Sequences
Two properties of a dynamical system, rigidity and non-recurrence, are
examined in detail. The ultimate aim is to characterize the sequences along
which these properties do or do not occur for different classes of
transformations. The main focus in this article is to characterize explicitly
the structural properties of sequences which can be rigidity sequences or
non-recurrent sequences for some weakly mixing dynamical system. For ergodic
transformations generally and for weakly mixing transformations in particular
there are both parallels and distinctions between the class of rigid sequences
and the class of non-recurrent sequences. A variety of classes of sequences
with various properties are considered showing the complicated and rich
structure of rigid and non-recurrent sequences
Perfect powers in polynomial power sums
We prove that a non-degenerate simple linear recurrence sequence of polynomials satisfying some further conditions
cannot contain arbitrary large powers of polynomials if the order of the
sequence is at least two. In other words we will show that for large
enough there is no polynomial of degree such that is an element of . The bound for
depends here only on the sequence . In the binary
case we prove even more. We show that then there is a bound on the index
of the sequence such that only elements with
index can be a proper power.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1810.1214
Non-recurrence sets for weakly mixing linear dynamical systems
We study non-recurrence sets for weakly mixing dynamical systems by using
linear dynamical systems. These are systems consisting of a bounded linear
operator acting on a separable complex Banach space X, which becomes a
probability space when endowed with a non-degenerate Gaussian measure. We
generalize some recent results of Bergelson, del Junco, Lema\'nczyk and
Rosenblatt, and show in particular that sets \{n_k\} such that n_{k+1}/{n_k}
tends to infinity, or such that n_{k} divides n_{k+1} for each k, are
non-recurrence sets for weakly mixing linear dynamical systems. We also give
examples, for each r, of r-Bohr sets which are non-recurrence sets for some
weakly mixing systems
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