90 research outputs found
Computing necessary integrability conditions for planar parametrized homogeneous potentials
Let V\in\mathbb{Q}(i)(\a_1,\dots,\a_n)(\q_1,\q_2) be a rationally
parametrized planar homogeneous potential of homogeneity degree . We design an algorithm that computes polynomial \emph{necessary} conditions
on the parameters (\a_1,\dots,\a_n) such that the dynamical system associated
to the potential is integrable. These conditions originate from those of
the Morales-Ramis-Sim\'o integrability criterion near all Darboux points. The
implementation of the algorithm allows to treat applications that were out of
reach before, for instance concerning the non-integrability of polynomial
potentials up to degree . Another striking application is the first complete
proof of the non-integrability of the \emph{collinear three body problem}.Comment: ISSAC'14 - International Symposium on Symbolic and Algebraic
Computation (2014
Monomization of Power Ideals and Parking Functions
In this note, we find a monomization of a certain power ideal associated to a
directed graph. This power ideal has been studied in several settings. The
combinatorial method described here extends earlier work of other, and will
work on several other types of power ideals, as will appear in later work
Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves
We show that for a class of two-loop diagrams, the on-shell part of the
integration-by-parts (IBP) relations correspond to exact meromorphic one-forms
on algebraic curves. Since it is easy to find such exact meromorphic one-forms
from algebraic geometry, this idea provides a new highly efficient algorithm
for integral reduction. We demonstrate the power of this method via several
complicated two-loop diagrams with internal massive legs. No explicit elliptic
or hyperelliptic integral computation is needed for our method.Comment: minor changes: more references adde
Markov bases and structural zeros
AbstractIn this paper we apply the elimination technique to the computation of Markov bases, paying special attention to contingency tables with structural zeros. An algebraic relationship between the Markov basis for a table with structural zeros and the corresponding complete table is proved. In order to find the relevant Markov basis, it is enough to eliminate the indeterminates associated with the structural zeros from the toric ideal for the complete table. Moreover, we use this result for the computation of Markov bases for some classical log-linear models, such as quasi-independence and quasi-symmetry, and computations in the multi-way setting are presented
On the complexity of computing Gr\"obner bases for weighted homogeneous systems
Solving polynomial systems arising from applications is frequently made
easier by the structure of the systems. Weighted homogeneity (or
quasi-homogeneity) is one example of such a structure: given a system of
weights , -homogeneous polynomials are polynomials
which are homogeneous w.r.t the weighted degree
. Gr\"obner bases for weighted homogeneous systems can be
computed by adapting existing algorithms for homogeneous systems to the
weighted homogeneous case. We show that in this case, the complexity estimate
for Algorithm~\F5 \left(\binom{n+\dmax-1}{\dmax}^{\omega}\right) can be
divided by a factor . For zero-dimensional
systems, the complexity of Algorithm~\FGLM (where is the
number of solutions of the system) can be divided by the same factor
. Under genericity assumptions, for
zero-dimensional weighted homogeneous systems of -degree
, these complexity estimates are polynomial in the
weighted B\'ezout bound .
Furthermore, the maximum degree reached in a run of Algorithm \F5 is bounded by
the weighted Macaulay bound , and this bound is
sharp if we can order the weights so that . For overdetermined
semi-regular systems, estimates from the homogeneous case can be adapted to the
weighted case. We provide some experimental results based on systems arising
from a cryptography problem and from polynomial inversion problems. They show
that taking advantage of the weighted homogeneous structure yields substantial
speed-ups, and allows us to solve systems which were otherwise out of reach
Thomas Decomposition of Algebraic and Differential Systems
In this paper we consider disjoint decomposition of algebraic and non-linear
partial differential systems of equations and inequations into so-called simple
subsystems. We exploit Thomas decomposition ideas and develop them into a new
algorithm. For algebraic systems simplicity means triangularity, squarefreeness
and non-vanishing initials. For differential systems the algorithm provides not
only algebraic simplicity but also involutivity. The algorithm has been
implemented in Maple
Evaluation properties of invariant polynomials
AbstractA polynomial invariant under the action of a finite group can be rewritten using generators of the invariant ring. We investigate the complexity aspects of this rewriting process; we show that evaluation techniques enable one to reach a polynomial cost
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