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This paper gives the recursion formula for mixed multiplicities of maximal degrees with respect to joint reductions of ideals, which is one of important results in the mixed multiplicity theory. Using this result, we give consequences on the relationship between mixed multiplicities and the Hilbert-Samuel multiplicity under most essential assumptions, that is a substantial progress in the problem of expressing mixed multiplicities into the Hilbert-Samuel multiplicity., Comment: 23 pages. arXiv admin note: text overlap with arXiv:1803.09201

In the direction towards the question when mixed multiplicities are equal to the Hilbert-Samuel multiplicity of joint reductions, this paper not only generalizes [28, Theorem 3.1] that covers the Rees's theorem [13, Theorem 2.4], but also removes the hypothesis that joint reductions are systems of parameters in [28, Theorem 3.1]. The results of the paper seem to make the problem of expressing mixed multiplicities into the Hilbert-Samuel multiplicity of joint reductions become closer., Comment: 15 pages

The original mixed multiplicity theory considered the class of mixed multiplicities concerning the terms of highest total degree in the Hilbert polynomial. This paper defines a broader class of mixed multiplicities that concern the maximal terms in this polynomial, and gives many results, which are not only general but also more natural than many results in the original mixed multiplicity theory., Comment: 20 pages. arXiv admin note: text overlap with arXiv:1211.1238

This paper shows that mixed multiplicities and the multiplicity of Rees modules of good filtrations and that of their reductions are the same. As an application of this result, we obtain interesting results on mixed multiplicities and the multiplicity of Rees modules., Comment: 15 pages

This paper defines the Euler-Poincar\'{e} characteristic of joint reductions of ideals which concerns the maximal terms in the Hilbert polynomial; characterizes the positivity of mixed multiplicities in terms of minimal joint reductions; proves the additivity and other elementary properties for mixed multiplicities. The results of the paper together with the results of [17] seem to show a natural and nice picture of mixed multiplicities of maximal degrees., Comment: 22 pages

This paper investigates the relationship between multiplicities and the degree sequence of ideals in graded algebras, gives multiplicity equations of graded rings via the degree sequence of ideals, and characterizes mixed multiplicities and multiplicities of Rees rings in terms of the degree sequence of ideals. As an application, the paper answered an open question on the multiplicity of Rees rings., Comment: 19 pages

This paper builds fundamental perfect fields of positive characteristic and shows the structure of perfect fields that a field of positive characteristic is a perfect field if and only if it is an algebraic extension of a fundamental perfect field., Comment: 4 pages

The paper gives some criteria for partial sums of rational number sequences to be not rational functions and to be not algebraic functions. As an application, we study partial sums of some famous rational number sequences in mathematical analysis with relevance to number theory., Comment: 9 pages

In this paper, we give a simple definition of tangents to a curve in elementary geometry. From which, we characterize the existence of the tangent to a curve at a point., Comment: 6 pages, 2 figures

This paper defines mixed multiplicity systems; the Euler-Poincare characteristic and the mixed multiplicity symbol of $\mathbb{N}^d$-graded modules with respect to a mixed multiplicity system, and proves that the Euler-Poincare characteristic and the mixed multiplicity symbol of any mixed multiplicity system of the type $(k_1,\ldots,k_d)$ and the $(k_1,\ldots,k_d)$-difference of the Hilbert polynomial are the same. As an application, we get results for mixed multiplicities., Comment: 24 pages

This paper establishes mixed multiplicity formulas concerning the relationship between mixed multiplicities of modules and mixed multiplicities of rings via rank of modules.

This paper gives the additivity and reduction formulas for mixed multiplicities of multi-graded modules $M$ and mixed multiplicities of arbitrary ideals, and establishes the recursion formulas for the sum of all the mixed multiplicities of $M.$ As an application of these formulas we get the recursion formulas for the multiplicity of multi-graded Rees modules.

Let $(A, \frak m)$ be a noetherian local ring with maximal ideal $\frak{m}$ and infinite residue field $k = A/\frak{m}.$ Let $J$ be an $\frak m$-primary ideal, $I_1,...,I_s$ ideals of $A$, and $M$ a finitely generated $A$-module. In this paper, we interpret mixed multiplicities of $(I_1,..., I_s,J)$ with respect to $M$ as multiplicities of joint reductions of them. This generalizes the Rees's theorem on mixed multiplicity (Theorem 2.4). As an application we show that mixed multiplicities are also multiplicities of Rees's superficial sequences., Comment: 14 pages

Let S be a finitely generated standard multi-graded algebra over a Noetherian local ring A. This paper first expresses mixed multiplicities of S in term of Hilbert-Samuel multiplicity that explained the mixed multiplicities S as the Hilbert-Samuel multiplicities for quotient modules of S. As an application, we get formulas for the mixed multiplicities of ideals that covers the main result of Trung-Verma 2007 [TV]., Comment: 17 pages

The paper characterizes the multiplicity and the Cohen-Macaulayness of fiber cones of good filtrations in terms of minimal reductions of these filtrations., Comment: 16 pages

Let $S$ be a finitely generated standard multigraded algebra over an Artinian local ring $A$; $M$ a finitely generated multigraded $S$-module. This paper answers to the question when mixed multiplicities of $M$ are positive and characterizes them in terms of lengths of $A$-modules. As an application, we get results on mixed multiplicities of ideals (Proposition 4.4 and Theorem 4.5), and recover the results of Risler and Teissier [Te](see Remark 4.8), Trung and Verma [TV](see Remark 4.6)., Comment: 23 pages

This paper shows that the main result of Trung-Verma in 2007 [TV] only is an immediate consequence of an improvement version of [Theorem 3.4, Vi1] in 2000., Comment: 4 pages