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Boundary slices and the 𝒫‐Euler condition
Author(s) -
Ghiloni Riccardo
Publication year - 2007
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdm051
Subject(s) - mathematics , boundary (topology) , euler's formula , polynomial , euler characteristic , ring (chemistry) , algebraic number , integer (computer science) , class (philosophy) , function (biology) , characterization (materials science) , set (abstract data type) , relation (database) , rational function , operator (biology) , pure mathematics , discrete mathematics , combinatorics , mathematical analysis , computer science , biochemistry , chemistry , materials science , organic chemistry , repressor , database , artificial intelligence , evolutionary biology , transcription factor , gene , biology , programming language , nanotechnology
Let X ⊂ℝ n be a closed semialgebraic set, and let ̃( X ) be the ring obtained from the characteristic function of X by the operations+,−, * and the half link operator, and by the polynomial operations with rational coefficients which preserve finite formal sums of signs. McCrory and Parusiński proved that a necessary condition for X to be homeomorphic to a real algebraic set is that X is ‐Euler; that is, all the functions in ̃( X ) are integer‐valued. In this paper, we introduce a class of subsets of X , called boundary slices of X . We establish a relation between these subsets of X and the ‐Euler condition on X , and we give some applications of this relation. As a consequence, we infer that all the arc‐symmetric semialgebraic sets and all the real analytic sets are ‐Euler, answering affirmatively a question of Kurdyka, McCrory and Parusiński.