arXiv subjects

Olga Kashcheyeva

Publications and source records attributed to Olga Kashcheyeva.

6 recordsLinked to original sources

arXiv2024

Generating sequences of rank 1 valuations

We consider a rank 1 valuation $ν$ centered in a regular 3-dimensional local ring $R$. We assume that the residue field $k$ of $ν$ is contained in $R$. An algorithm for constructing a generating sequences for $ν$ in $R$ is provided. This is a generalization of our previous algorithm where only valuations of rational rank 1 were considered. We then give a comparison example of a rational rank 3 valuation centered in a ring of polynomials in three variables for which we explicitly compute a set of key polynomials and a generating sequence.

math.AC↗

arXiv2015

Constructing examples of semigroups of valuations

We work with rational rank 1 valuations centered in regular local rings. Given an algebraic function field $K$ of transcendence degree 3 over $k$, a regular local ring $R$ with $QF(R)=K$ and a $k$-valuation $ν$ of $K$, we provide an algorithm for constructing a generating sequences for $ν$ in $R$. We then develop a method for determining a valuation $ν$ on $k(x,y,z)$ through the sequence of defining values. Using the above results we construct examples of valuations centered in $k[x,y,z]_{(x,y,z)}$ and investigate their semigroups of values.

math.AC↗

arXiv2008

Growth of rank 1 valuation semigroups

We consider the question of which semigroups can occur as the semigroup $S_R(ν)$ of positive values of a rank 1 valuation dominating a Noetherian local ring $R$. We give a number of bounds of polynomial type on the growth of $ϕ(n)=S_R(ν)\cap (0,n)$ for $n\in\NN$, starting with the upper bound of $P_R(n)$, where $P_R(n)$ is the Hilbert function of $R$. This bound is generalized to an extremely general bound for arbitrary rank valuations in the paper "Semigroups of valuations on local rings, II", by Cutkosky and Teissier, arXiv:0805.3788. This bound is already enough to give simple examples of rank 1 well ordered semigroups which are not the value semigroup $S_R(ν)$ of a valuation dominating a Noetherian local ring. In the case of rank 1, it is possible to give more precise estimates of $ϕ(n)$, which we prove in this paper. We also give examples showing that many different rates of growth are possible for $ϕ(n)$ on a regular local ring of dimension 2, such as $n(α$ for any rational $α$ with $1\leα\le 2$, and $n{log}(n)$.

math.AC↗

arXiv2008

Algebraic series and valuation rings over nonclosed fields

Suppose that $k$ is an arbitrary field. Consider the field $k((x_1,...,x_n))$, which is the quotient field of the ring $k[[x_1,...,x_n]]$ of formal power series in the variables $x_1,...,x_n$, with coefficients in $k$. Suppose that $σ$ is a formal power series in $x_1,...,x_n$ with coefficints in the algebraic closure of $k$. We give a very simple necessary and sufficient condition for $σ$ to be algebraic over $k((x_1,...,x_n))$. As an application of our methods, we give a characterization of valuation rings $V$ which dominate an excellent, Noetherian local domain $R$ of dimension two, and such that the rank increases after passing to the completion of a birational extension of $R$.

math.AC↗

arXiv2007

Toroidalization of generating sequences in dimension two function fields of positive characteristic

We give a characteristic free proof of the main result of our previous paper (math.AC/0509697) concerning toroidalization of generating sequences of valuations in dimension two function fields. We show that when an extension of two dimensional algebraic regular local rings $R\subset S$ satisfies the conclusions of the Strong Monomialization theorem of Cutkosky and Piltant, the map between generating sequences in $R$ and $S$ has a toroidal structure.

math.AC↗

arXiv2005

Toroidalization of generating sequences in dimension two function fields

Let k be an algebraically closed field of characteristic 0 and let K*/K be a finite extension of algebraic function fields of transcendence degree 2 over k. Let v* be a k-valuation of K* with valuation ring V* and let v be the restriction of v* to K. Suppose R --> S is an extension of algebraic regular local rings with quotient fields K and K*, respectively, such that V* dominates S and S dominates R. We prove that there exist sequences of quadratic transforms R --> R' and S --> S' along v* such that S' dominates R' and the map between generating sequences of v and v* in R' and S', respectively, has a toroidal structure. Our result extends the Strong Monomialization theorem of Cutkosky and Piltant.

math.AC↗