Decompositions of Ideals of Minors Meeting a Submatrix

Ideals of Adjacent Minors

We give a description of the minimal primes of the ideal generated by the 2×2 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent m×m minors of an m×n generic matrix when the characteristic of the ground field is zero. A key intermediate result is the proof that the ideals which appear as minimal primes are, in fact, prime ideals. This ...

Decompositions of Binomial Ideals

A binomial ideal in a polynomial ring is an ideal which can be generated by binomials. We present Binomials, a package for the computer algebra system Macaulay 2 (Eisenbud et al (2001)), which specializes well known algorithms to binomial ideals, allowing for a significant speedup of common computations like primary decomposition. Central parts of the implemented algorithms go back to the found...

Simplicial Decompositions, Tree-decompositions and Graph Minors

The concepts of simplicial decompositions, tree-decompositions and simplicial tree-decompositions were all inspired by a common forerunner: the decompositions of finite graphs used by K. Wagner in his classic paper [ 13 ], in which he proved the equivalence of the 4-Colour-Conjecture to Hadwiger’s Conjecture for n = 5. To show that the 4CC implies Hadwiger’s Conjecture (for n = 5), Wagner used ...

Optimal shadows and ideals in submatrix orders

We consider the poset of all submatrices of a given matrix, ordered by containment. The unique rank function for this poset is given by r(M) = R(M)+C(M)?1, where R(M) and C(M) denote the number of rows and columns of a nonempty matrix M, respectively, and the rank of the empty matrix is 0. For xed k and i our objective is to nd a set M of submatrices M 1 ; M 2 ; : : :; M k such that r(M j) = i ...

Bidiagonal decompositions, minors and applications