Abstract
We address the decision problem for sentences involving univariate functions constructed from a fixed Pfaffian function of order 1. We present a new symbolic procedure solving this problem with a computable complexity based on the computation of suitable Sturm sequences. For a general Pfaffian function, we assume the existence of an oracle to determine the sign that a function of the class takes at a real algebraic number. As a by-product, we obtain a new oracle-free effective algorithm solving the same problem for univariate E-polynomials based on techniques that are simpler than the previous ones, and we apply it to solve a similar decision problem in the multivariate setting. Finally, we introduce a notion of Thom encoding for zeros of an E-polynomial and describe an algorithm for their computation.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Khovanskii, A.: On a class of systems of transcendental equations. Soviet Math. Dokl. 22, 762–765 (1980)
Khovanskii, A.: Fewnomials. Translations of Mathematical Monographs, 88. American Mathematical Society, Providence, RI, (1991)
Gabrielov, A., Vorobjov, N.: Complexity of computations with Pfaffian and Noetherian functions. In Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, Kluwer (2004)
Tarski, A.: A decision method for elementary algebra and geometry, 2nd edn. Universtiy of California Press, Berkeley, Los Angeles (1951)
van den Dries, L.: Remarks on Tarski’s problem concerning \(({\mathbf{R}},+,\cdot ,\text{exp})\). Logic Colloquium ’82 (Florence, 1982), 97–121, Stud. Logic Found. Math., 112, North-Holland, Amsterdam (1984)
Macintyre, A., Wilkie, A.J.: On the decidability of the real exponential field. Kreiseliana, 441–467, A K Peters, Wellesley, MA (1996)
Richardson, D.: Towards computing non algebraic cylindrical decompositions. In: Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation (ISSAC '91), pp. 247–255. Association for Computing Machinery, New York, NY, USA (1991)
Vorobjov, N.: The complexity of deciding consistency of systems of polynomials in exponent inequalities. J. Symbolic Comput. 13(2), 139–173 (1992)
Maignan, A.: Solving one and two-dimensional exponential polynomial systems. In: Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation (ISSAC '98), pp. 215–221. Association for Computing Machinery, New York, NY, USA (1998)
Anai, H., Weispfenning, V.: Deciding linear-trigonometric problems. In: Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation (ISSAC '00), pp. 14–22. Association for Computing Machinery, New York, NY, USA (2000)
Weispfenning, V.: Deciding linear-transcendental problems. In: Computer Algebra in Scientific Computing, pp. 423–437. Springer, Berlin (2000)
Achatz, M., McCallum, S., Weispfenning, V.: Deciding polynomial-exponential problems. In: Proceedings of the 2008 International Symposium on Symbolic and Algebraic Computation (ISSAC '08), pp. 215–222. Association for Computing Machinery, New York, NY, USA (2008)
Roy, M.-F., Vorobjov, N.: Finding irreducible components of some real transcendental varieties. Comput. Complexity 4, 107–132 (1994)
McCallum, S., Weispfenning, V.: Deciding polynomial-transcendental problems. J. Symbol. Comput. 47, 16–31 (2013)
Xu, M., Li, Z.-B., Yang, L.: Quantifier elimination for a class of exponential polynomial formulas. J. Symbolic Comput. 68, 146–168 (2015)
Barbagallo, M.L., Jeronimo, G., Sabia, J.: Zero counting for a class of univariate Pfaffian functions. J. Algebra 452, 549–573 (2016)
van der Hoeven, J., Shackell, J.R.: Complexity bounds for zero-test algorithms. J. Symbolic Comput. 41, 1004–1020 (2006)
Waldschmidt, M.: Transcendence measures for exponentials and logarithms. J. Austral. Math. Soc. Ser. A 25(4), 445–465 (1978)
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003)
Basu, S., Pollack, R., Roy, M.-F.: Algorithms in real algebraic geometry. Second edition. Algorithms and Computation in Mathematics, 10. Springer-Verlag, Berlin, (2006). Online version available at http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted3.html
Perrucci, D.: Linear solving for sign determination. Theoret. Comput. Sci. 412(35), 4715–4720 (2011)
Heindel, L.E.: Integer arithmetic algorithms for polynomial real zero determination. J. Assoc. Comput. Mach. 18, 533–548 (1971)
Canny, J.: Improved algorithms for sign determination and existencial quantifier elimination. Computer Science Division, University of California, Berkeley 94720 USA. Comput. J. Vol. 36, No 5, (1993)
Acknowledgements
The authors wish to thank the anonymous referee for their helpful comments and suggestions which contributed to improve the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Partially supported by the following Grants: PIP 11220130100527CO (CONICET), UBACYT 20020160100039BA (2017-2020) and UBACYT 20020190100116BA (2020-2022).
Rights and permissions
About this article
Cite this article
Barbagallo, M.L., Jeronimo, G. & Sabia, J. Decision problem for a class of univariate Pfaffian functions. AAECC 35, 207–232 (2024). https://doi.org/10.1007/s00200-022-00545-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-022-00545-8