Let R be a real closed field with a real valuation v. A ℤ-valued semialgebraic function on Rⁿ is called algebraic if it can be written as the sign of a symmetric bilinear form over R[X₁,. .., Xₙ]. We show that the reduction of such a function with respect to v is again algebraic on the residue field. This implies a corresponding result for limits of algebraic functions in definable families.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-3,
author = {Ludwig Br\"ocker},
title = {Reduction of semialgebraic constructible functions},
journal = {Annales Polonici Mathematici},
volume = {85},
year = {2005},
pages = {27-38},
zbl = {1091.14015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-3}
}
Ludwig Bröcker. Reduction of semialgebraic constructible functions. Annales Polonici Mathematici, Tome 85 (2005) pp. 27-38. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-3/