Integration of semialgebraic functions and integrated Nash functions

Abstract

When integrating semialgebraic functions one has to leave the semialgebraic setting. For example, one gets the global logarithm and iterated antiderivatives of algebraic power series such as the arctangent. We show that it is enough to enlarge the semialgebraic functions by these functions to completely describe parameterized integrals of semialgebraic functions. To realize this we close the rings of algebraic power series in arbitrary dimension under taking antiderivatives. We analyze these rings profoundly. In particular we show that the Weierstrass division theorem and the Weierstrass preparation theorem hold. This allows us to apply model theoretic results to obtain an explicit description of parameterized integrals of semialgebraic functions. Finally, we investigate the structure generated by the integrated algebraic power series.