Decision problem for a class of univariate Pfaffian functions
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 funct...
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| Published in: | Applicable algebra in engineering, communication and computing Vol. 35; no. 2; pp. 207 - 232 |
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| Language: | English |
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01.03.2024
Springer Nature B.V |
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| ISSN: | 0938-1279, 1432-0622 |
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| 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. |
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| AbstractList | 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. |
| Author | Jeronimo, Gabriela Sabia, Juan Barbagallo, María Laura |
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| Cites_doi | 10.1016/j.tcs.2011.05.006 10.1016/j.jsc.2006.06.001 10.1525/9780520348097 10.1145/120694.120732 10.1145/321662.321667 10.1007/BF01202285 10.1016/S0747-7171(08)80088-0 10.1017/S1446788700021431 10.1145/345542.345567 10.1145/281508.281616 10.1016/j.jalgebra.2015.11.050 10.1007/978-3-642-57201-2_32 10.1016/j.jsc.2011.08.004 10.1016/j.jsc.2014.09.015 10.1007/3-540-33099-2 10.1090/mmono/088 10.1007/978-94-007-1025-2_5 10.1016/S0049-237X(08)71811-1 10.1093/comjnl/36.5.409 |
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| Keywords | 68W30 Pfaffian functions 14P15 Decision problem Sturm sequences Complexity |
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| SubjectTerms | Algebra Algorithms Artificial Intelligence Computation Computer Hardware Computer Science Estimates Original Paper Partial differential equations Polynomials Sequences Symbolic and Algebraic Manipulation Theory of Computation |
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