52 research outputs found
Near NP-Completeness for Detecting p-adic Rational Roots in One Variable
We show that deciding whether a sparse univariate polynomial has a p-adic
rational root can be done in NP for most inputs. We also prove a
polynomial-time upper bound for trinomials with suitably generic p-adic Newton
polygon. We thus improve the best previous complexity upper bound of EXPTIME.
We also prove an unconditional complexity lower bound of NP-hardness with
respect to randomized reductions for general univariate polynomials. The best
previous lower bound assumed an unproved hypothesis on the distribution of
primes in arithmetic progression. We also discuss how our results complement
analogous results over the real numbers.Comment: 8 pages in 2 column format, 1 illustration. Submitted to a conferenc
Counting Value Sets: Algorithm and Complexity
Let be a prime. Given a polynomial in \F_{p^m}[x] of degree over
the finite field \F_{p^m}, one can view it as a map from \F_{p^m} to
\F_{p^m}, and examine the image of this map, also known as the value set. In
this paper, we present the first non-trivial algorithm and the first complexity
result on computing the cardinality of this value set. We show an elementary
connection between this cardinality and the number of points on a family of
varieties in affine space. We then apply Lauder and Wan's -adic
point-counting algorithm to count these points, resulting in a non-trivial
algorithm for calculating the cardinality of the value set. The running time of
our algorithm is . In particular, this is a polynomial time
algorithm for fixed if is reasonably small. We also show that the
problem is #P-hard when the polynomial is given in a sparse representation,
, and is allowed to vary, or when the polynomial is given as a
straight-line program, and is allowed to vary. Additionally, we prove
that it is NP-hard to decide whether a polynomial represented by a
straight-line program has a root in a prime-order finite field, thus resolving
an open problem proposed by Kaltofen and Koiran in
\cite{Kaltofen03,KaltofenKo05}
Ultrametric Fewnomial Theory
An ultrametric field is a field that is locally compact as a metric space with
respect to a non-archimedean absolute value. The main topic of this dissertation is
to study roots of polynomials over such fields.
If we have a univariate polynomial with coefficients in an ultrametric field and
non-vanishing discriminant, then there is a bijection between the set of roots of the
polynomial and classes of roots of the same polynomial in a finite ring. As a consequence,
there is a ball in the polynomial space where all polynomials in it have the
same number of roots.
If a univariate polynomial satisfies certain generic conditions, then we can efficiently
compute the exact number of roots in the field. We do that by using Hensel's
lemma and some properties of Newton's polygon.
In the multivariate case, if we have a square system of polynomials, we consider
the tropical set which is the intersection of the tropical varieties of its polynomials.
The tropical set contains the set of valuations of the roots, and for every point in
the tropical set, there is a corresponding system of lower polynomials. If the system
satisfies some generic conditions, then for each point w in the tropical set the number
of roots of valuation w equals the number roots of valuation w of the lower system.
The last result enables us to compute the exact number of roots of a polynomial
system where the tropical set is finite and the lower system consists of binomials.
This algorithmic method can be performed in polynomial-time if we fix the number of variables.
We conclude the dissertation with a discussion of the feasibility problem. We
consider the problem of the p-adic feasibility of polynomials with integral coefficients
with the prime number p as a part of the input. We prove this problem can be solved
in nondeterministic polynomial-time. Furthermore, we show that any problem, which
can be solved in nondeterministic polynomial-time, can be reduced to this feasibility
problem in randomized polynomial-time
Bounded-degree factors of lacunary multivariate polynomials
In this paper, we present a new method for computing bounded-degree factors
of lacunary multivariate polynomials. In particular for polynomials over number
fields, we give a new algorithm that takes as input a multivariate polynomial f
in lacunary representation and a degree bound d and computes the irreducible
factors of degree at most d of f in time polynomial in the lacunary size of f
and in d. Our algorithm, which is valid for any field of zero characteristic,
is based on a new gap theorem that enables reducing the problem to several
instances of (a) the univariate case and (b) low-degree multivariate
factorization.
The reduction algorithms we propose are elementary in that they only
manipulate the exponent vectors of the input polynomial. The proof of
correctness and the complexity bounds rely on the Newton polytope of the
polynomial, where the underlying valued field consists of Puiseux series in a
single variable.Comment: 31 pages; Long version of arXiv:1401.4720 with simplified proof
Quantitative Aspects of Sums of Squares and Sparse Polynomial Systems
Computational algebraic geometry is the study of roots of polynomials and polynomial systems. We are familiar with the notion of degree, but there are other ways to consider a polynomial: How many variables does it have? How many terms does it have? Considering the sparsity of a polynomial means we pay special attention to the number of terms. One can sometimes profit greatly by making use of sparsity when doing computations by utilizing tools from linear programming and integer matrix factorization. This thesis investigates several problems from the point of view of sparsity. Consider a system F of n polynomials over n variables, with a total of n + k distinct exponent vectors over any local field L. We discuss conjecturally tight bounds on the maximal number of non-degenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, and L only. In particular, we give new explicit systems with number of roots approaching the best known upper bounds. We also give a complete classification for when an n-variate n + 2-nomial positive polynomial can be written as a sum of squares of polynomials. Finally, we investigate the problem of approximating roots of polynomials from the viewpoint of sparsity by developing a method of approximating roots for binomial systems that runs more efficiently than other current methods. These results serve as building blocks for proving results for less sparse polynomial systems
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