11 research outputs found
On the complexity of computing with zero-dimensional triangular sets
We study the complexity of some fundamental operations for triangular sets in
dimension zero. Using Las-Vegas algorithms, we prove that one can perform such
operations as change of order, equiprojectable decomposition, or quasi-inverse
computation with a cost that is essentially that of modular composition. Over
an abstract field, this leads to a subquadratic cost (with respect to the
degree of the underlying algebraic set). Over a finite field, in a boolean RAM
model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm
for modular composition. Conversely, we also show how to reduce the problem of
modular composition to change of order for triangular sets, so that all these
problems are essentially equivalent. Our algorithms are implemented in Maple;
we present some experimental results
Computing Puiseux series : a fast divide and conquer algorithm
Let be a polynomial of total degree defined over
a perfect field of characteristic zero or greater than .
Assuming separable with respect to , we provide an algorithm that
computes the singular parts of all Puiseux series of above in less
than operations in , where
is the valuation of the resultant of and its partial derivative with
respect to . To this aim, we use a divide and conquer strategy and replace
univariate factorization by dynamic evaluation. As a first main corollary, we
compute the irreducible factors of in up to an
arbitrary precision with arithmetic
operations. As a second main corollary, we compute the genus of the plane curve
defined by with arithmetic operations and, if
, with bit operations
using a probabilistic algorithm, where is the logarithmic heigth of .Comment: 27 pages, 2 figure
On the complexity of computing real radicals of polynomial systems
International audienceLet f= (f1, ..., fs) be a sequence of polynomials in Q[X1,...,Xn] of maximal degree D and V⊂ Cn be the algebraic set defined by f and r be its dimension. The real radical re associated to f is the largest ideal which defines the real trace of V . When V is smooth, we show that re , has a finite set of generators with degrees bounded by V. Moreover, we present a probabilistic algorithm of complexity (snDn )O(1) to compute the minimal primes of re . When V is not smooth, we give a probabilistic algorithm of complexity sO(1) (nD)O(nr2r) to compute rational parametrizations for all irreducible components of the real algebraic set V ∩ Rn. Experiments are given to show the efficiency of our approaches
Computing critical points for invariant algebraic systems
Let be a field and , in
be multivariate polynomials (with )
invariant under the action of , the group of permutations of
. We consider the problem of computing the points at which
vanish and the Jacobian matrix associated to is
rank deficient provided that this set is finite. We exploit the invariance
properties of the input to split the solution space according to the orbits of
. This allows us to design an algorithm which gives a triangular
description of the solution space and which runs in time polynomial in ,
and where is the maximum degree of the
input polynomials. When are fixed, this is polynomial in while when
is fixed and this yields an exponential speed-up with respect
to the usual polynomial system solving algorithms
Computing Puiseux series: a fast divide and conquer algorithm
Let be a polynomial of total degree D defined over a field K of characteristic zero or greater than D. Assuming F separable with respect to Y , we provide an algorithm that computes all Puiseux series of F above X = 0 in less than operations in K, where δ is the valuation of the resultant of F and its partial derivative with respect to Y. To this aim, we use a divide and conquer strategy and replace univariate factorisation by dynamic evaluation. As a first main corollary, we compute the irreducible factors of F in up to an arbitrary precision X N with arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with arithmetic operations and, if K = Q, with bit operations using probabilistic algorithms, where h is the logarithmic height of F
Homotopy algorithms for solving structured determinantal systems
Multivariate polynomial systems arising in numerous applications have special structures. In particular, determinantal structures and invariant systems appear in a wide range of applications such as in polynomial optimization and related questions in real algebraic geometry. The goal of this thesis is to provide efficient algorithms to solve such structured systems.
In order to solve the first kind of systems, we design efficient algorithms by using the symbolic homotopy continuation techniques. While the homotopy methods, in both numeric and symbolic, are well-understood and widely used in polynomial system solving for square systems, the use of these methods to solve over-detemined systems is not so clear. Meanwhile, determinantal systems are over-determined with more equations than unknowns. We provide probabilistic homotopy algorithms which take advantage of the determinantal structure to compute isolated points in the zero-sets of determinantal systems. The runtimes of our algorithms are polynomial in the sum of the multiplicities of isolated points and the degree of the homotopy curve. We also give the bounds on the
number of isolated points that we have to compute in three contexts: all entries of the input are in classical polynomial rings, all these polynomials are sparse, and they are weighted polynomials.
In the second half of the thesis, we deal with the problem of finding critical points of a symmetric polynomial map on an invariant algebraic set. We exploit the invariance properties of the input to split the solution space according to the orbits of the symmetric group. This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in the number of points that we have to compute. Our results are illustrated by applications in studying real algebraic sets defined by invariant polynomial systems by the means of the critical point method