Jump of Milnor numbers

In this note we study a problem of A'Campo about the minimal non-zero difference between the Milnor numbers of a germ of plane curve and one of its deformation.     

An elementary coordinate-dependent local resolution of singularities and applications

An elementary classical analysis resolution of singularities method is developed, extensively using explicit coordinate systems. The algorithm is designed to be applicable to subjects such as oscillatory integrals and critical integrability exponents. As one might expect, the trade-off for such an elementary method is a weaker theorem than Hironaka's work [H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, Ann. of Math. (2) 79 (1964) 109-203; H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero II, Ann. of Math. (2) 79 (1964) 205-326] or its subsequent simplications and extensions such as [E. Bierstone, P. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (2) (1997) 207-302; S. Encinas, O. Villamayor, Good points and constructive resolution of singularities, Acta Math. 181 (1) (1998) 109-158; J. Kollar, Resolution of singularities—Seattle lectures, preprint; A.N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Funct. Anal. Appl. 18 (3) (1976) 175-196]. Nonetheless the methods of this paper can be used to prove a variety of theorems of interest in analysis. As illustration, two consequences are given. First and most notably, a general theorem regarding the existence of critical integrability exponents are established. Secondly, a quick proof of a well-known inequality of Lojasiewicz [S. Lojasiewicz, Ensembles semi-analytiques, Inst. Hautes Études Sci., Bures-sur-Yvette, 1964] is given. The arguments here are substantially different from the general algorithms such as [H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, Ann. of Math. (2) 79 (1964) 109-203; H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero II, Ann. of Math. (2) 79 (1964) 205-326], or the elementary arguments of [E. Bierstone, P. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988) 5-42] and [H. Sussman, Real analytic desingularization and subanalytic sets: an elementary approach, Trans. Amer. Math. Soc. 317 (2) (1990) 417-461]. The methods here have as antecedents the earlier work of the author [M. Greenblatt, A direct resolution of singularities for functions of two variables with applications to analysis, J. Anal. Math. 92 (2004) 233-257], Phong and Stein [D.H. Phong, E.M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997) 107-152], and Varchenko [A.N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Funct. Anal. Appl. 18 (3) (1976) 175-196].     

Every place admits local uniformization in a finite extension of the function field

We prove that every place P of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F of F. We show that F|F can be chosen to be Galois, after a finite purely inseparable extension of the ground field K. Instead of being Galois, the extension can also be chosen such that the induced extension FP|FP of the residue fields is purely inseparable and the value group of F only gets divided by the residue characteristic. If F lies in the completion of an Abhyankar place, then no extension of F is needed. Our proofs are based solely on valuation theoretical theorems, which are of particular importance in positive characteristic. They are also applicable when working over a subring R⊂K and yield similar results if R is regular and of dimension smaller than 3.     

On a conjecture of Wan about limiting Newton polygons

We show that for a monic polynomial $f(x)$ over a number field K containing a global permutation polynomial of degree >1 as its composition factor, the Newton Polygon of $f\mathrm{mod}\mathfrak{p}$ does not converge for $\mathfrak{p}$ passing through all finite places of K. In the rational number field case, our result is the “only if” part of a conjecture of Wan about limiting Newton polygons.     

Ramification of valuations

In this paper we develop a very explicit theory of ramification of general valuations in algebraic function fields. In characteristic zero and arbitrary dimension, we obtain the strongest possible generalization of the classical ramification theory of local Dedekind domains. We further develop a ramification theory of algebraic functions fields of dimension two in positive characteristic. We prove that local monomialization and simultaneous resolution hold under very mild assumptions, and give pathological examples.     

Clemens-Schmid序列的推广

本文把Clemens-Schmid正合序列推广到单参数族的奇异纤维具有任意奇性的情况。     

More congruences for numerical data of an embedded resolution

To an arbitrary intersection of exceptional varieties of an embedded resolution we associate a finite number of congruences between naturally occurring multiplicities. This theory generalizes previous results concerning just one exceptional variety. Moreover we describe precise equalities which imply the congruences and we give some applications on the poles of Igusa's local zeta function.     

A Family of Algebraically Closed Fields Containing Polynomials in Several Variables

We introduce a family of fields of series with support in strongly convex rational cones. All these fields contain polynomials in several variables. We prove that they are algebraically closed with a construction that is analogous to the Newton polygon for algebraic curves. As a corollary, we show the existence of fractional power solutions with support in cones for systems of equations.     

一类解析的零级Laplace-Stieltjes变换

本文研究了Laplace-Stieltjes变换所定义的零级整函数的增长性.利用牛顿多边形,得到了这类整函数关于增长性及正规增长性的充要条件,推广了文献[3]中的结果.     

Schönemann–Eisenstein–Dumas-Type Irreducibility Conditions that Use Arbitrarily Many Prime Numbers

The famous irreducibility criteria of Schönemann–Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper, we will use some results and ideas of Dumas to provide several irreducibility criteria of Schönemann–Eisenstein–Dumas-type for polynomials with integer coefficients, criteria that are given by some divisibility conditions for their coefficients with respect to arbitrarily many prime numbers. A special attention will be paid to those irreducibility criteria that require information on the divisibility of the coefficients by two distinct prime numbers.     

On L-functions of certain exponential sums

Let ${\mathbb{F}}_q$ denote the finite field of order q of characteristic p. We study the p-adic valuations for zeros of L-functions associated with exponential sums of the following family of Laurent polynomials$f(x)=a_1{x}_{n+1}(x_1+\frac{1}{x_1})+\cdots +a_n{x}_{n+1}(x_n+\frac{1}{x_n})+a_{n+1}{x}_{n+1}+\frac{1}{x_{n+1}}$ where $a_i\in {\mathbb{F}}_q^{⁎}$, $i=1,2,\dots ,n+1$. When $n=2$, the estimate of the associated exponential sum appears in Iwaniecʼs work on small eigenvalues of the Laplace–Beltrami operator acting on automorphic functions with respect to the group ${\Gamma }_0(p)$, and Adolphson and Sperber gave complex absolute values for zeros of the corresponding L-function. Using the decomposition theory of Wan, we determine the generic Newton polygon (q-adic values of the reciprocal zeros) of the L-function. Working on the chain level version of Dworkʼs trace formula and using Wanʼs decomposition theory, we are able to give an explicit Hasse polynomial for the generic Newton polygon in low dimensions, i.e., $n⩽3$.     

全平面上解析的零级Laplace-Stieltjes变换

本文研究了Laplace-Stieltjes变换所定义的零级整函数的增长性.利用型函数,得到了这类整函数关于增长性及正规增长性的充要条件,推广了Dirichlet级数的相关结论.     

Singularities on complete algebraic varieties

We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety. The first author was partially supported by NSF grant DMS-01591 and the third author was partially supported by NSF grant DMS-03693