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 Aisequivalenttoastatementaboutanarbitrarypairofpoints\alpha,\beta\in\sper\ Aandtheirseparatingideal;werefertothisstatementastheLocalPierce−Birkhoffconjectureat\alpha,\beta.Inthispaper,foreachpair(\alpha,\beta)withht()=\dim A,wedefineanaturalnumber,calledcomplexityof(\alpha,\beta).Complexity0correspondstothecasewhenoneofthepoints\alpha,\betaismonomial;thiscasewasalreadysettledinalldimensionsinaprecedingpaper.Hereweintroduceanewconjecture,calledtheStrongConnectednessconjecture,andprovethatthestrongconnectednessconjectureindimensionn−1impliestheconnectednessconjectureindimensionninthecasewhenht()islessthann−1.WeprovetheStrongConnectednessconjectureindimension2,whichgivestheConnectednessandthePierce−−Birkhoffconjecturesinanydimensioninthecasewhenht()lessthan2.Finally,weprovetheConnectedness(andhencealsothePierce−−Birkhoff)conjectureinthecasewhendimensionofAisequaltoht()=3,thepair(\alpha,\beta)isofcomplexity1andA$ is excellent with residue field the field of real numbers
Let R be a real closed field and A=R[x_1,...,x_n]. Let sper A denote the real
spectrum of A. There are two kinds of points in sper A : finite points (those
for which all of |x_1|,...,|x_n| are bounded above by some constant in R) and
points at infinity. In this paper we study the structure of the set of points
at infinity of sper A and their associated valuations. Let T be a subset of
{1,...,n}. For j in {1,...,n}, let y_j=x_j if j is not in T and y_j=1/x_j if j
is in T. Let B_T=R[y_1,...,y_n]. We express sper A as a disjoint union of sets
of the form U_T and construct a homeomorphism of each of the sets U_T with a
subspace of the space of finite points of sper B_T. For each point d at
infinity in U_T, we describe the associated valuation v_{d*} of its image d* in
sper B_T in terms of the valuation v_d associated to d. Among other things we
show that the valuation v_{d*} is composed with v_d (in other words, the
valuation ring R_d is a localization of R_{d*} at a suitable prime ideal)
7 pagesHistorical overview of Nash Problem of arcs in the EMS NewsletterThe goal of this paper is to give a historical overview of the Nash Problem of arcs in arbitrary dimension, as well as its a rmative solution in dimension two by J. Fernandez de Bobadilla and M. Pe Pereira and a negative solution in higher dimensions by T. de Fernex, S. Ishii and J. Koll ar. This problem was stated by J. Nash around 1963 and has been an important subject of research in singularity theory
The Casas-Alvero conjecture predicts that every univariate polynomial over a
field of characteristic zero having a common factor with each of its
derivatives Hi​(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 dpâ„“,
ℓ∈N, where p is a good prime for d. In this paper we
calculate certain distinguished monomials appearing in the resultant
R(f,Hi​(f)) and obtain a (non-exhaustive) list of bad primes for every degree
d∈N∖{0}