Abstract
It is known that every germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that the homeomorphism can be chosen in such a way that the analytic and algebraic germs are tangent with any prescribed order of tangency. Moreover, the space of arcs contained in the algebraic germ approximates the space of arcs contained in the analytic one, in the sense that they are identical up to a prescribed truncation order.
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Notes
A pseudopolynomial is a polynomial in \(x_i\) with coefficients that are analytic in the other variables. The pseudopolynomials \(F_i\) that we consider are moreover distinguished polynomials in x, i.e. are of the form \(\displaystyle x_i^p+\sum _{j=1}^pa_j(x^{i-1})x_i^{p-j}\) with \(a_j(0)=0\) for all j. They may depend analytically on t that is considered as a parameter.
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The authors were partially supported by ANR project STAAVF (ANR-2011 BS01 009). G. Rond was partially supported by ANR project SUSI (ANR-12-JS01-0002-01). M. Bilski was partially supported by the NCN Grant 2014/13/B/ST1/00543.
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Bilski, M., Kurdyka, K., Parusiński, A. et al. Higher order approximation of analytic sets by topologically equivalent algebraic sets. Math. Z. 288, 1361–1375 (2018). https://doi.org/10.1007/s00209-017-1937-5
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DOI: https://doi.org/10.1007/s00209-017-1937-5