1 S-polynomials eliminating the leading term Buchberger's criterion and algorithm 2 Wavelet Design construct wavelet filters 3 Proof of the Buchberger Criterion two lemmas proof of the Buchberger criterion termination and elimination

Gröbner Bases와 응용

A complex hypersurface D in C-n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields We classify all LFDs for n at most 4
 
By analogy with Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of C-n \ D We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields is reductive For n at most 4, we show that the GLCT holds for all LFDs
 
We show that LFDs arising naturally as discriminants in quiver representation spaces (of real Schur roots) fulfill the GLCT As a by-product we obtain a topological proof of a theorem of V Kac on the number of irreducible components of such discriminants

/pdf/linear-free-divisors-and-the-global-logarithmic-comparison-3clsnlzmxv.pdf

Linear free divisors and the global logarithmic comparison theorem

An algorithm of computing $b$-functions

Boolean Gröbner bases

Gröbner bases and the number of Latin squares related to autotopisms of order ≤7

Sudoku is a logic-based placement puzzle. We recall how to translate this puzzle into a 9-colouring problem which is equivalent to a (big) algebraic system of polynomial equations. We study how far Grobner bases techniques can be used to treat these systems produced by Sudokus. This general purpose tool can not. be considered as a good solver, but we show that it can be useful to provide information on systems that are -in spite of their origin- - hard to solve.

Sudokus and gröbner bases : Not only a Divertimento

Let D, x be a free divisor germ in a complex manifold X of dimension n > 2. It is an open problem to find out which are the properties required for D,x to satisfy the so-called Logarithmic Comparison Theorem (LCT), that is, when the complex of logarithmic differential forms computes the cohomology of the complement of D,x. We give a family of Euler homogeneous free divisors which, somewhat unexpectedly, does not satisfy the LCT.

/pdf/logarithmic-comparison-theorem-and-some-euler-homogeneous-55q0mvz66h.pdf

Logarithmic Comparison Theorem and some Euler homogeneous free divisors

Sudoku is a logic-based placement puzzle. We recall how to translate this puzzle into a 9-colouring problem which is equivalent to a (big) algebraic system of polynomial equations. We study how far Grobner bases techniques can be used to treat these systems produced by Sudokus. This general purpose tool can not be considered as a good solver, but we show that it can be useful to provide information on systems that are —in spite of their origin— hard to solve.

/pdf/sudokus-and-grobner-bases-not-only-a-divertimento-34zcbd1282.pdf

José Maria Ucha-Enriquez

Papers

Sudokus and gröbner bases : Not only a Divertimento

Logarithmic Comparison Theorem and some Euler homogeneous free divisors

Sudokus and Gröbner bases: not only a divertimento

Comparison of theoretical complexities of two methods for computing annihilating ideals of polynomials

Gröbner bases and logarithmic D-modules