Papers by Mark Spivakovsky
Journal of Pure and Applied Algebra, Mar 1, 2023
Let ν be a rank one valuation on K[x] and Ψn the set of key polynomials for ν of degree n ∈ N. We... more Let ν be a rank one valuation on K[x] and Ψn the set of key polynomials for ν of degree n ∈ N. We discuss the concepts of being Ψn-stable and (Ψn, Q)-fixed. We discuss when these two concepts coincide. We use this discussion to present a simple proof of Proposition 8.2 of [3] and Theorem 1.2 of .
Michigan Mathematical Journal, Jun 1, 2017
This is a continuation of a previous paper by the same authors. In the former paper, it was prove... more This is a continuation of a previous paper by the same authors. In the former paper, it was proved that in order to obtain local uniformization for valuations centered on local domains, it is enough to prove it for rank one valuations. In this paper, we extend this result to the case of valuations centered on rings which are not necessarily integral domains and may even contain nilpotents.
Springer eBooks, 2020
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Sandwiched surface singularities and the Nash resolution for surfaces
arXiv (Cornell University), Jul 5, 2023
The main goal of this paper is to characterize the module of Kähler differentials for an extensio... more The main goal of this paper is to characterize the module of Kähler differentials for an extension of valuation rings. More precisely, we consider a simple algebraic valued field extension (L/K, v) and the corresponding valuation rings O L and O K . In the case when e(L/K, v) = 1 we present a characterization for Ω O L /O K in terms of a given sequence of key polynomials for the extension. Moreover, we use our main result to present a characterization for when Ω O L /O K = {0}.
arXiv (Cornell University), Jul 27, 2012
This paper represents a step in our program towards the proof of the Pierce-Birkhoff conjecture. ... more This paper represents a step in our program towards the proof of the Pierce-Birkhoff conjecture. In the nineteen eighties J. Madden proved that the Pierce-Birkhoff conjecture for a ring A is equivalent to a statement about an arbitrary pair of points α, β ∈ Sper A and their separating ideal < α, β >; we refer to this statement as the local Pierce-Birkhoff conjecture at α, β. In we introduced a slightly stronger conjecture, also stated for a pair of points α, β ∈ Sper A and the separating ideal < α, β >, called the Connectedness conjecture. In this paper, for each pair (α, β) with ht(< α, β >) = dim A, we define a natural number, called complexity of (α, β). Complexity 0 corresponds to the case when one of the points α, β is monomial; this case was settled in all dimensions in . In the present paper we introduce a new conjecture, called the Strong Connectedness conjecture, and prove that the strong connectedness conjecture in dimension n -1 implies the connectedness conjecture in dimension n in the case when ht(< α, β >) ≤ n -1. We prove the Strong Connectedness conjecture in dimension 2, which gives the Connectedness and the Pierce-Birkhoff conjectures in any dimension in the case when ht(< α, β >) ≤ 2. Finally, we prove the Connectedness (and hence also the Pierce-Birkhoff) conjecture in the case when dim A = ht(< α, β >) = 3, the pair (α, β) is of complexity 1 and A is excellent with residue field R.
Local uniformization in characteristic zero. Archimedean case
HAL (Le Centre pour la Communication Scientifique Directe), 2008
International audienc
Sandwiched Surface Singularities And the Nash Resolution Problem
Advanced studies in pure mathematics, Jun 6, 2018
International Mathematics Research Notices, May 28, 2016
We explore some equisingularity criteria in one parameter families of generically reduced curves.... more We explore some equisingularity criteria in one parameter families of generically reduced curves. We prove the equivalence between Whitney regularity and Zariski's discriminant criterion. We prove that topological triviality implies smoothness of the normalized surface. Examples are given to show that Witney regularity and equisaturation are not stable under the blow-up of the singular locus nor under the Nash modification.
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, Feb 9, 2013
Part 2. Higher rank and higher dimensional valuations 37 8. Higher rank valuations 37 9. Higher d... more Part 2. Higher rank and higher dimensional valuations 37 8. Higher rank valuations 37 9. Higher dimensional valuations 37 Part 3. Globalization 38 References 40 Proof. By Hironaka's reduction of the singularities (see ) of M 0 , we get a nonsingular projective model M ′ of K jointly with a birational morphism M ′ → M 0 that is the composition of a finite sequence of blow-ups with non-singular centers. Consider the local ring O M ′ ,P ′ of M ′ at the center P ′ of ν and chose elements f 1 , f 2 , . . . , f r ∈ O M,P such that ν(f 1 ), ν(f 2 ), . . . , ν(f r ) are Z-linearly independent. Another application of Hironaka's theorem gives a birational morphism M → M ′ , that is also a composition of a finite sequence of blow-ups with non-singular centers, such thatf = r i=1 f i , is a monomial (times a unit) in a suitable regular system
arXiv (Cornell University), Nov 17, 2016
In this paper we introduce a new concept of key polynomials for a given valuation ν on K[x]. We p... more In this paper we introduce a new concept of key polynomials for a given valuation ν on K[x]. We prove that such polynomials have many of the expected properties of key polynomials as those defined by MacLane and Vaquié, for instance, that they are irreducible and that the truncation of ν associated to each key polynomial is a valuation. Moreover, we prove that every valuation ν on K[x] admits a sequence of key polynomials that completely determines ν (in the sense which we make precise in the paper). We also establish the relation between these key polynomials and pseudoconvergent sequences defined by Kaplansky.
arXiv (Cornell University), Sep 21, 2015
In this paper we give a short introduction to the local uniformization problem. This follows a si... more In this paper we give a short introduction to the local uniformization problem. This follows a similar line as the one presented by the second author in his talk at ALANT 3. We also discuss our paper on the reduction of local uniformization to the rank one case. In that paper, we prove that in order to obtain local uniformization for valuations centered at objects of a subcategory of the category of noetherian integral domains, it is enough to prove it for rank one valuations centered at objects of the same category. We also announce an extension of this work which was partially developed during ALANT 3. This extension says that the reduction mentioned above also works for noetherian rings with zero divisors (including the case of non-reduced rings).
Some results on quasi-unmixed local domains
Journal of Algebra, Sep 1, 2015
ABSTRACT This paper shows that if is a Nœtherian unibranch local domain with field of fractions K... more ABSTRACT This paper shows that if is a Nœtherian unibranch local domain with field of fractions K, then the integral closure S of R in K is analytically irreducible and finite over R if and only if R is analytically irreducible. We also prove the equality between the number of minimal prime ideals in and the number of maximal ideals in S in the case when R is a Nœtherian quasi-unmixed local domain such that S is finite over R and has only one minimal prime for all the maximal ideals in S, where is the -adic completion of .
HAL (Le Centre pour la Communication Scientifique Directe), 2009
Let R be a real closed field. The Pierce-Birkhoff conjecture says that any piecewise polynomial f... more Let R be a real closed field. The Pierce-Birkhoff conjecture says that any piecewise polynomial function f on R n can be obtained from the polynomial ring R[x 1 , . . . , x n ] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points α, β ∈ Sper R[x 1 , . . . , x n ], the existence of connected sets in the real spectrum of R[x 1 , . . . , x n ], satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce-Birkhoff conjecture. Secondly, we construct a class of connected sets in the real spectrum which, though not in itself enough for the proof of the Pierce-Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes the first step in our program for a proof of the Pierce-Birkhoff conjecture in dimension greater than 2. Thirdly, we apply these ideas to give two proofs of the Connectedness conjecture (and hence also of the Pierce-Birkhoff conjecture in the abstract formulation) in the special case when one of the two points α, β ∈ Sper R[x 1 , . . . , x n ] is monomial. One of the proofs is elementary while the other consists in deducing the (monomial) Connectedness conjecture as an immediate corollary of the main connectedness theorem of this paper.
Mark Spivakovsky Interview July 15, 1982
NOTE: to view these items please visit http://dynkincollection.library.cornell.eduInterview condu... more NOTE: to view these items please visit http://dynkincollection.library.cornell.eduInterview conducted by Eugene Dynkin with Mark Spivakovsky in Durham, New Hampshire on July 15, 1982 at the AMS Summer Research Conference "Quantum fields, Probability and Geometry," July 11-17, 1982
A counterexample to the theorem of Beppo Levi in three dimensions
Inventiones Mathematicae, Feb 1, 1989
... IJl ve~l tiolle$ mathematicae 9 Springer-Verlag 1989 A counterexample to the theorem of Beppo... more ... IJl ve~l tiolle$ mathematicae 9 Springer-Verlag 1989 A counterexample to the theorem of Beppo Levi in three dimensions Mark Spivakovsky* Department of Mathematics, Harvard University, Cambridge, Ma 02138, USA ... 519-521). Theorem of Beppo Levi ([4, 5], p. 522). ...
Publications of The Research Institute for Mathematical Sciences, 1982
Valuations in Function Fields of Surfaces
American Journal of Mathematics, Feb 1, 1990
... In fact, this definition makes sense for any D such that the I-adic order in CX,D is a valuat... more ... In fact, this definition makes sense for any D such that the I-adic order in CX,D is a valuation. ... Abhyankar's proof uses the structure theorem for complete regular local rings and a comparison theorem of valuations in a local ring with those in its completion [1, Proposition 5. See ...
arXiv (Cornell University), Jul 12, 2023
The Casas-Alvero conjecture predicts that every univariate polynomial over a field of characteris... more The Casas-Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives H i (f ) is a power of a linear polynomial. One approach to proving the conjecture is to first prove it for polynomials of some small degree d, compile a list of bad primes for that degree (namely, those primes p for which the conjecture fails in degree d and characteristic p) and then deduce the conjecture for all degrees of the form dp ℓ , ℓ ∈ N, where p is a good prime for d. In this paper we calculate certain distinguished monomials appearing in the resultant R(f, H i (f )) and obtain a (non-exhaustive) list of bad primes for every degree d ∈ N \ {0}.
HAL (Le Centre pour la Communication Scientifique Directe), Nov 27, 2022
In this paper, we study the structure of the graded ring associated to a limit key polynomial Qn ... more In this paper, we study the structure of the graded ring associated to a limit key polynomial Qn in terms of the key polynomials that define Qn. In order to do that, we use direct limits. In general, we describe the direct limit of a family of graded rings associated to a totally ordered set of valuations. As an example, we describe the graded ring associated to a valuation-algebraic valuation as a direct limit of graded rings associated to residue-transcendental valuations.
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Papers by Mark Spivakovsky