Section: New Results
Fundamental algorithms and structured polynomial systems
Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences
The so-called Berlekamp – Massey – Sakata algorithm
computes a Gröbner
basis of a 0-dimensional ideal of relations satisfied by an input
table. It extends the Berlekamp – Massey algorithm
to
In the extended version [6], we investigate this problem and design several algorithms for computing such a Gröbner basis of an ideal of relations using linear algebra techniques. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations.
As each query to the table can be expensive, we design a second algorithm requiring fewer queries, in general. This FGLM -like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a multi-Hankel matrix (a multivariate generalization of Hankel matrices).
Under some
additional assumptions, we make a third, adaptive, algorithm and reduce
further the number of table queries.
Then, we relate the number of queries of
this third algorithm to the
geometry of the final staircase and we show that it is
essentially linear in the size of the output when the staircase is convex.
As a direct application to this, we decode
We show that the multi-Hankel matrices are heavily structured when using the LEX ordering and that we can speed up the computations using fast algorithms for quasi-Hankel matrices. Finally, we design algorithms for computing the generating series of a linear recursive table.
Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra
Given several
A P-recursive sequence
Finally, we show how to incorporate Gröbner bases computations in an
Ore algebra
On the Connection Between Ritt Characteristic Sets and Buchberger-Gröbner Bases
For any polynomial ideal
On the complexity of computing Gröbner 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
Gröbner bases for weighted homogeneous systems can be computed by
adapting existing algorithms for homogeneous systems to the weighted
homogeneous case. In [12], we show that in
this case, the complexity estimate for Algorithm F5
Furthermore, the maximum degree reached in a run of Algorithm F5 is
bounded by the weighted Macaulay bound
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.
A Superfast Randomized Algorithm to Decompose Binary Forms
Symmetric Tensor Decomposition is a major problem that arises in areas such as signal processing, statistics, data analysis and computational neuroscience. It is equivalent to a homogeneous polynomial in
On the Bit Complexity of Solving Bilinear Polynomial Systems
In [29] we bound the Boolean complexity of
computing isolating hyperboxes
for all complex roots of systems of bilinear polynomials.
The resultant of such systems admits a family of determinantal
Sylvester-type formulas, which we make explicit by means of
homological complexes.
The computation of the determinant of the resultant matrix
is a bottleneck for the overall complexity. We exploit the
quasi-Toeplitz structure to reduce the problem to efficient
matrix-vector products, corresponding to multivariate polynomial
multiplication.
For zero-dimensional systems, we arrive at a primitive element
and a rational univariate
representation of the roots. The overall bit complexity of our
probabilistic algorithm is