23 research outputs found
Explicit formula for the generating series of diagonal 3D rook paths
Let denote the number of ways in which a chess rook can move from a
corner cell to the opposite corner cell of an
three-dimensional chessboard, assuming that the piece moves closer to the goal
cell at each step. We describe the computer-driven \emph{discovery and proof}
of the fact that the generating series admits
the following explicit expression in terms of a Gaussian hypergeometric
function: G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27
w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.Comment: To appear in "S\'eminaire Lotharingien de Combinatoire
Counting and computing regions of -decomposition: algebro-geometric approach
New methods for -decomposition analysis are presented. They are based on
topology of real algebraic varieties and computational real algebraic geometry.
The estimate of number of root invariant regions for polynomial parametric
families of polynomial and matrices is given. For the case of two parametric
family more sharp estimate is proven. Theoretic results are supported by
various numerical simulations that show higher precision of presented methods
with respect to traditional ones. The presented methods are inherently global
and could be applied for studying -decomposition for the space of parameters
as a whole instead of some prescribed regions. For symbolic computations the
Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure
On the computation of -flat outputs for differential-delay systems
We introduce a new definition of -flatness for linear differential delay
systems with time-varying coefficients. We characterize - and -0-flat
outputs and provide an algorithm to efficiently compute such outputs. We
present an academic example of motion planning to discuss the pertinence of the
approach.Comment: Minor corrections to fit with the journal versio
On the complexity of skew arithmetic
13 pagesIn this paper, we study the complexity of several basic operations on linear differential operators with polynomial coefficients. As in the case of ordinary polynomials, we show that these complexities can be expressed in terms of the cost of multiplication
Automatic Classification of Restricted Lattice Walks
International audienceWe propose an leading to the computer-driven of various conjectures about structural properties of generating functions coming from enumeration of restricted lattice walks in 2D and in 3D
Counting Solutions of a Polynomial System Locally and Exactly
We propose a symbolic-numeric algorithm to count the number of solutions of a
polynomial system within a local region. More specifically, given a
zero-dimensional system , with
, and a polydisc
, our method aims to certify the existence
of solutions (counted with multiplicity) within the polydisc.
In case of success, it yields the correct result under guarantee. Otherwise,
no information is given. However, we show that our algorithm always succeeds if
is sufficiently small and well-isolating for a -fold
solution of the system.
Our analysis of the algorithm further yields a bound on the size of the
polydisc for which our algorithm succeeds under guarantee. This bound depends
on local parameters such as the size and multiplicity of as well
as the distances between and all other solutions. Efficiency of
our method stems from the fact that we reduce the problem of counting the roots
in of the original system to the problem of solving a
truncated system of degree . In particular, if the multiplicity of
is small compared to the total degrees of the polynomials ,
our method considerably improves upon known complete and certified methods.
For the special case of a bivariate system, we report on an implementation of
our algorithm, and show experimentally that our algorithm leads to a
significant improvement, when integrated as inclusion predicate into an
elimination method
Sharper and Simpler Nonlinear Interpolants for Program Verification
Interpolation of jointly infeasible predicates plays important roles in
various program verification techniques such as invariant synthesis and CEGAR.
Intrigued by the recent result by Dai et al.\ that combines real algebraic
geometry and SDP optimization in synthesis of polynomial interpolants, the
current paper contributes its enhancement that yields sharper and simpler
interpolants. The enhancement is made possible by: theoretical observations in
real algebraic geometry; and our continued fraction-based algorithm that rounds
off (potentially erroneous) numerical solutions of SDP solvers. Experiment
results support our tool's effectiveness; we also demonstrate the benefit of
sharp and simple interpolants in program verification examples