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.
Similar content being viewed by others
References
Bauer, H.: Measure and Integration theory. de Gruyter Studies in Mathematics. Walter de Gruyter, Germany (2001)
Bochnak, J., Coste, M., Roy, M.-F.: Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete 36, Springer, Heidelberg (1998)
Cartan, H.: Differential forms. Translated from the French. Houghton Mifflin Co., Boston (1970)
Cluckers, R., Miller, D.: Stability under integration of sums of products of real globally subanalytic functions and their logarithms. Duke Math. J. 156(2), 311–348 (2011)
Comte, G., Lion, J.-M., Rolin, J.-P.: Nature log-analytique du volume des sous-analytiques. Illinois J. Math. 44(4), 884–888 (2000)
Denef, J., Lipshitz, L.: Ultraproducts and approximation in local rings. II. Math. Ann. 253(1), 1–28 (1980)
van den Dries, L.: On the elementary theory of restricted elementary functions. J. Symbolic Logic 53(3), 796–808 (1988)
van den Dries, L., Miller, C.: Extending Tamm’s theorem. Ann. Inst. Fourier 44(5), 1367–1395 (1994)
Gunning, R., Rossi, H.: Analytic functions of several complex variables. Reprint of the original AMS Chelsea Publishing, Providence, RI (1965) 2009
Kaiser, T.: First order tameness of measures. Ann. Pure Appl. Logic 163(12), 1903–1927 (2012)
Kontsevich, M., Zagier, D.: Periods. Mathematics unlimited: and beyond, pp. 771–808. Springer, Berlin (2001)
Lion, J.-M., Rolin, J.-P.: Théorème de préparation pour les fonctions logarithmico-exponentielles. Ann. Inst. Fourier 47(3), 859–884 (1997)
Lion, J.-M., Rolin, J.-P.: Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques. Ann. Inst. Fourier 48(3), 755–767 (1998)
Miller, D.: A preparation theorem for Weierstrass systems. Trans. Amer. Math. Soc. 358(10), 4395–4439 (2006)
Ruiz, J.: The basic theory of power series. Advanced lectures in mathematics. Vieweg, Germany (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was supported in part by DFG KA 3297/1-1.
Rights and permissions
About this article
Cite this article
Kaiser, T. Integration of semialgebraic functions and integrated Nash functions. Math. Z. 275, 349–366 (2013). https://doi.org/10.1007/s00209-012-1138-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-012-1138-1