Mathematics

We build on the correspondence between abstract key polynomials and minimal pairs made by Novacoski and show how to relate the valuations that are generated by each object. We can then give a geometric interpretation of valuations built in this fashion. To do so we employ an object called diskoid, which is a generalisation of the classical concept of ball in non-archimedian valued fields.

We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a rational map via a computation of the multiplicity of the saturated special fiber ring. As specific applications, we provide explicit formulas for all the projective degrees of rational maps determined by perfect ideals of height two or by Gorenstein ideals of height three.

By double ideal quotient, we mean (I:(I:J)) where ideals I and J . In our previous work [11], double ideal quotient and its variants are shown to be very useful for checking prime divisor and generating primary component. Combining those properties, we can compute "direct localization" effectively, comparing with full primary decomposition. In this paper, we apply modular techniques effectively to computation of such double ideal quotient and its variants, where first we compute them modulo several prime numbers and then lift them up over rational numbers by Chinese Remainder Theorem and rational reconstruction. As a new modular technique for double ideal quotient and its variants, we devise criteria for output from modular computations. Also, we apply modular techniques to intermediate primary decomposition. We examine the effectiveness of our modular techniques for several examples by preliminary computational experiences on Singular.

We prove that the multi-Rees algebra R( I 1 ?�⋯??I r ) of a collection of strongly stable ideals I 1 ,?? I r is of fiber type. In particular, we provide a Gröbner basis for its defining ideal as a union of a Gröbner basis for its special fiber and binomial syzygies. We also study the Koszulness of R( I 1 ?�⋯??I r ) based on parameters associated to the collection. Furthermore, we establish a quadratic Gröbner basis of the defining ideal of R( I 1 ??I 2 ) where each of the strongly stable ideals has two quadric Borel generators. As a consequence, we conclude that this multi-Rees algebra is Koszul.

V is a complete intersection scheme in a multiprojective space if it can be defined by an ideal I with as many generators as codim(V) . We investigate the multigraded regularity of complete intersections scheme in P n ? P m . We explicitly compute many values of the Hilbert functions of 0 -dimensional complete intersections. We show that these values only depend upon n,m , and the bidegrees of the generators of I . As a result, we provide a sharp upper bound for the multigraded regularity of 0 -dimensional complete intersections.

We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of m -primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of r filtrations on an analytically unramified local ring R come from the coefficients of a suitable homogeneous polynomial in r variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of m -primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterization of equality in the Minkowski inequality for m-primary ideals in a local ring by Teissier, Rees and Sharp and Katz does not extend to arbitrary filtrations, but we show that they are true in a large and important subcategory of filtrations. We define divisorial and bounded filtrations. The filtration of powers of a fixed ideal is a bounded filtration, as is a divisorial filtration. We show that in an excellent local domain, the characterization of equality in the Minkowski equality is characterized by the condition that the integral closures of suitable Rees like algebras are the same, strictly generalizing the theorem of Teissier, Rees and Sharp and Katz. We also prove that a theorem of Rees characterizing the inclusion of ideals with the same multiplicity generalizes to bounded filtrations in excellent local domains. We give a number of other applications, extending classical theorems for ideals.

We prove that two arbitrary ideals I⊂J in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. We also obtain a Principle of Specialization of Integral Dependence, which gives a condition for integral dependence in terms of the constancy of the multiplicity sequence in families.

Given a graph G whose edges are labeled by ideals of a commutative ring R with identity, a generalized spline is a vertex labeling of G by the elements of R so that the difference of labels on adjacent vertices is an element of the corresponding edge ideal. The set of all generalized splines on a graph G with base ring R has a ring and an R -module structure. In this paper, we focus on the freeness of generalized spline modules over certain graphs with the base ring R=k[ x 1 ,?? x d ] where k is a field. We first show the freeness of generalized spline modules on graphs with no interior edges over k[x,y] such as cycles or a disjoint union of cycles with free edges. Later, we consider graphs that can be decomposed into disjoint cycles without changing the isomorphism class of the syzygy modules. Then we use this decomposition to show that generalized spline modules are free over k[x,y] and later we extend this result to the base ring R=k[ x 1 ,?? x d ] under some restrictions.

In this paper, we introduce the concept of N-pure ideal as a generalization of pure ideal. Using this concept, a new and interesting type of rings is presented, we call it mid ring. Also, we provide new characterizations for von Neumann regular and zero-dimensional rings. Moreover, some results about mp-ring are given. Finally, a characterization for mid rings is provided. Then it is shown that the class of mid rings is strictly between the class of reduced mp-rings (p.f. rings) and the class of mp-rings.

Exploiting the interior-closure duality developed by Epstein and R.G., we show that for the class of Matlis dualizable modules M over a Noetherian local ring, when cl is a Nakayama closure and i its dual interior, there is a duality between cl-reductions and i-expansions that leads to a duality between the cl-core of modules in M and the i-hull of modules in M ∨ . We further show that many algebra and module closures and interiors are Nakayama and describe a method to compute the interior of ideals using closures and colons. We use our methods to give a unified proof of the equivalence of F-rationality with F-regularity, and of F-injectivity with F-purity, in the complete Gorenstein local case. Additionally, we give a new characterization of the finitistic tight closure test ideal in terms of maps from R 1/ p e . Moreover, we show that the liftable integral spread of a module exists.