Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a cross-section is adde...

We will study monomial ideals I in the exterior algebra as well as in the polynomial ring whose generic initial ideal is constant for all term orders up to permutations of variables. First, in the exterior algebra, we determine all graphs and all flag complexes whose exterior face ideal satisfies the above condition. Second, in the polynomial ring, it will be shown that the generic initial idea...

The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal o...

Given a vertex-weighted oriented graph, we can associate to it set of monomials. We consider the toric ideal whose defining map is given by these find generating for certain classes graphs which depends on combinatorial structure and weights graph. provide result analogous unweighted, unoriented graph case, show that when associated simple has only trivial even closed walks, zero ideal. Moreove...

Let G be a finite simple graph with edge ideal I (G). Let I (G)∨ denote the Alexander dual of I (G). We show that a description of all induced cycles of odd length in G is encoded in the associated primes of (I (G)∨)2. This result forms the basis for a method to detect odd induced cycles of a graph via ideal operations, e.g., intersections, products and colon operations. Moreover, we get a simp...

Let G denote a finite simple graph with edge ideal I(G). Letting I(G) denote the Alexander dual of I(G), we show that a description of the induced cycles of G of odd length is encoded in the associated primes of (I(G)). This result forms the basis for an algorithm to detect all the odd induced cycles of a graph via ideal operations, e.g., intersections, products, and colon operations. Moreover,...

The major results of Robertson and Seymour on graph well-quasi-ordering establish nonconstructively that many natural graph properties that constitute ideals in the minor or immersion orders are characterized by a nite set of forbidden substructures termed the obstructions for the property. This raises the question of what general kinds of information about an ideal are su cient, or insu cient,...