Sparse FGLM algorithms

@article{Faugre2017SparseFA,
  title={Sparse FGLM algorithms},
  author={Jean-Charles Faug{\`e}re and Chenqi Mou},
  journal={J. Symb. Comput.},
  year={2017},
  volume={80},
  pages={538-569}
}

Figures and Tables from this paper

Block-Krylov techniques in the context of sparse-FGLM algorithms

Finer Complexity Estimates for the Change of Ordering of Gröbner Bases for Generic Symmetric Determinantal Ideals

TLDR
This paper focuses on the Sparse-FGLM algorithm, the state-of-the-art for changing ordering of Gröbner bases of zero-dimensional ideals, and studies its complexity for symmetric determinantal ideals under a variant of Fröberg's conjecture.

Sparse FGLM using the block Wiedemann algorithm

TLDR
This work restricts its attention to systems with finitely many solutions, and typically first computes a Gröbner basis for the degrevlex ordering, and then converts it to either a lex GröBner basis or a related representation, such as Rouillier's Rational Univariate Representation.

Real root finding for low rank linear matrices

TLDR
Computer algebra algorithms, based on advanced methods for polynomial system solving, are designed to solve the problem of finding elements of low rank in a real affine subspace of dimension n efficiently and exactly.

Efficient linear algebra on GPUs for Gröbner bases computations

TLDR
This work aims to compute a lexicographic Gröbner basis of the ideal spanned by f1, which generically is of the form gn = 0, xn−1 = gn−1 (xn), . . . , x1 = g1 ( xn), deggn = D.

Faster Change of Order Algorithm for Gröbner Bases under Shape and Stability Assumptions

TLDR
The Hermite normal form of that matrix yields the sought lexicographic Gröbner basis, under assumptions which cover the shape position case, which improves upon both state-of-the-art complexity bounds O~(tD2) and O ~(Dω, since ω<3 and t≤D), and confirms the high practical benefit.

Solving Sparse Polynomial Systems using Gröbner Bases and Resultants

TLDR
This work will review these classical tools, their extensions, and recent progress in exploiting sparsity for solving polynomial systems, such as resultant computations, homotopy continuation methods, and most recently, Gröbner bases.

Exact algorithms for linear matrix inequalities

TLDR
An exact algorithm is designed that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty.

Guessing Gr{ö}bner Bases of Structured Ideals of Relations of Sequences

References

SHOWING 1-10 OF 55 REFERENCES

Fast algorithm for change of ordering of zero-dimensional Gröbner bases with sparse multiplication matrices

TLDR
This work investigates the particular but important shape position case and obtains an implementation which is able to manipulate 0-dimensional ideals over a prime field of degree greater than 30000 and outperforms the Magma/Singular/FGb implementations of FGLM.

Sub-cubic change of ordering for Gröbner basis: a probabilistic approach

TLDR
A new Las Vegas algorithm is proposed for solving polynomial systems with a finite number of solutions by using Gröbner basis for which the change of ordering step has a sub-cubic complexity and whose total complexity is dominated by the complexity of the F5 algorithm.

Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations

TLDR
This paper gives upper and lower bounds for the degrees of the elements of a Gr6bner base based on projective algebraic geometry and develops all the relations between these algorithms related to Gaussian elimination.

Using Symmetries in the Index Calculus for Elliptic Curves Discrete Logarithm

TLDR
This paper shows how to take advantage of some symmetries of twisted Edwards and twisted Jacobi intersections curves to gain an exponential factor 2ω(n−1) to solve the corresponding PDP where ω is the exponent in the complexity of multiplying two dense matrices.

Computing loci of rank defects of linear matrices using Gröbner bases and applications to cryptology

TLDR
The theoretical and practical complexity of computing Gröbner bases of two algebraic formulations of the MinRank problem are given and the determinantal ideal formulation is used to break a cryptographic challenge and allow us to evaluate precisely the security of the cryptosystem w.r.t. n, r and k.

On the Decoding of Cyclic Codes Using Gröbner Bases

TLDR
An algorithm for decoding cyclic codes up to their true minimum distance using Gröbner basis techniques is revisited and a geometric characterization of the number of errors is given, and the corresponding algebraic characterization is analyzed.

Critical points and Gröbner bases: the unmixed case

TLDR
The first complexity estimates on the computation of Gröbner bases of systems defining critical points are provided, showing the complexity is polynomial in the generic number of critical points, i.e. the number of variables in n-1/p-1 and exponential in p-i.

Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

TLDR
An FGLM-like algorithm for finding the relations in the table is produced, which lets us use linear algebra techniques and make use of fast structured linear algebra similarly to the Hankel interpretation of Berlekamp--Massey.
...