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We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power series of bounded complexity lying on an algebraic variety defined over the field of power series.
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Sets of zero-dimensional ideals in the polynomial ring $k[x,y]$ that share the same leading term ideal with respect to a given term ordering are known to be affine spaces called Grobner cells. Conca-Valla and Constantinescu parametrize such Grobner cells in terms of certain canonical Hilbert-Burch matrices for the lexicographical and degree-lexicographical term orderings, respectively. In this paper, we give a parametrization of $(x,y)$-primary ideals in Grobner cells which is compatible with the local structure of such ideals. More precisely, we extend previous results to the local setting by defining a notion of canonical Hilbert-Burch matrices of zero-dimensional ideals in the power series ring $k[[x,y]]$ with a given leading term ideal with respect to a local term ordering.
We consider Ribenboims construction of rings of generalized power series. Ribenboims construction makes use of a special class of partially ordered monoids and a special class of their subsets. While the restrictions he imposes might seem conceptually unclear, we demonstrate that they are precisely the appropriate conditions to represent such monoids as internal monoids in an appropriate category of Ehrhards finiteness spaces. Ehrhard introduced finiteness spaces as the objects of a categorical model of classical linear logic, where a set is equipped with a class of subsets to be thought of as finitary. Morphisms are relations preserving the finitary structure. The notion of finitary subset allows for a sharper analysis of computational structure than is available in the relational model. For example, fixed point operators fail to be finitary. In the present work, we take morphisms to be partial functions preserving the finitary structure rather than relations. The resulting category is symmetric monoidal closed, complete and cocomplete. Any pair of an internal monoid in this category and a ring induces a ring of generalized power series by an extension of the Ribenboim construction based on Ehrhards notion of linearization of a finiteness space. We thus further generalize Ribenboims constructions. We give several examples of rings which arise from this construction, including the ring of Puiseux series and the ring of formal power series generated by a free monoid.
Let fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{fa}(M), the finiteness dimension of M with respect to fa, and, its dual notion q_{fa}(M), the Artinianess dimension of M with respect to fa. When (R,fm) is local and r:=f_{fa}(M) is less than f_{fa}^{fm}(M), the fm-finiteness dimension of M relative to fa, we prove that H^r_{fa}(M) is not Artinian, and so the filter depth of fa on M doesnt exceeds f_{fa}(M). Also, we show that if M has finite dimension and H^i_{fa}(M) is Artinian for all i>t, where t is a given positive integer, then H^t_{fa}(M)/fa H^t_{fa}(M) is Artinian. It immediately implies that if q:=q_{fa}(M)>0, then H^q_{fa}(M) is not finitely generated, and so f_{fa}(M)leq q_{fa}(M).
Our goal is to settle a fading problem, the Jacobian Conjecture $(JC_n)$~: If $f_1, cdots, f_n$ are elements in a polynomial ring $k[X_1, cdots, X_n]$ over a field $k$ of characteristic zero such that $ det(partial f_i/ partial X_j) $ is a nonzero constant, then $k[f_1, cdots, f_n] = k[X_1, cdots, X_n]$. Practically, what we deal with is the generalized one, oindent The Generalized Jacobian Conjecture$(GJC)$ :{it Let $S hookrightarrow T$ be an unramified homomorphism of Noetherian domains. Assume that $S$ is a simply connected UFD ({sl i.e.,} ${rm Spec}(S)$ is simply connected and $S$ is a unique factorization domain) and that $T^times cap S = S^times$. Then $T = S$.} In addition, for consistency of the discussion, we raise some serious (or idiot) questions and some comments about the examples appeared in the papers published by the certain excellent mathematicians (though we are not willing to deal with them). However, the existence of such examples would be against our Main Result above, so that we have to dispute in Appendix B their arguments about the existence of their respective (so called) counter-examples. Our conclusion is that they are not perfect counter-examples which is shown explicitly.
This paper is devoted to the study of multigraded algebras and multigraded linear series. For an $mathbb{N}^s$-graded algebra $A$, we define and study its volume function $F_A:mathbb{N}_+^sto mathbb{R}$, which computes the asymptotics of the Hilbert function of $A$. We relate the volume function $F_A$ to the volume of the fibers of the global Newton-Okounkov body $Delta(A)$ of $A$. Unlike the classical case of standard multigraded algebras, the volume function $F_A$ is not a polynomial in general. However, in the case when the algebra $A$ has a decomposable grading, we show that the volume function $F_A$ is a polynomial with non-negative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series. Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties.
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