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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| Vydáno v: | Applicable algebra in engineering, communication and computing Ročník 35; číslo 2; s. 207 - 232 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Springer Berlin Heidelberg
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 |
| Author_xml | – sequence: 1 givenname: María Laura surname: Barbagallo fullname: Barbagallo, María Laura organization: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciclo Básico Común, Departamento de Ciencias Exactas, Universidad de Buenos Aires – sequence: 2 givenname: Gabriela surname: Jeronimo fullname: Jeronimo, Gabriela organization: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciclo Básico Común, Departamento de Ciencias Exactas, Universidad de Buenos Aires, Instituto de Investigaciones Matemáticas ”Luis A. Santaló” (IMAS), Consejo Nacional de Investigaciones Científicas y Técnicas, Universidad de Buenos Aires – sequence: 3 givenname: Juan surname: Sabia fullname: Sabia, Juan email: jsabia@dm.uba.ar organization: Ciclo Básico Común, Departamento de Ciencias Exactas, Universidad de Buenos Aires, Instituto de Investigaciones Matemáticas ”Luis A. Santaló” (IMAS), Consejo Nacional de Investigaciones Científicas y Técnicas, Universidad de Buenos Aires |
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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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| References | CR2 Waldschmidt (CR18) 1978; 25 CR3 CR6 McCallum, Weispfenning (CR14) 2013; 47 CR5 Roy, Vorobjov (CR13) 1994; 4 Barbagallo, Jeronimo, Sabia (CR16) 2016; 452 Richardson (CR7) 1991 Tarski (CR4) 1951 van der Hoeven, Shackell (CR17) 2006; 41 CR23 Weispfenning (CR11) 2000 CR20 Xu, Li, Yang (CR15) 2015; 68 Perrucci (CR21) 2011; 412 Vorobjov (CR8) 1992; 13 Achatz, McCallum, Weispfenning (CR12) 2008 Khovanskii (CR1) 1980; 22 Maignan (CR9) 1998 Heindel (CR22) 1971; 18 Anai, Weispfenning (CR10) 2000 von zur Gathen, Gerhard (CR19) 2003 545_CR5 545_CR6 S McCallum (545_CR14) 2013; 47 M Xu (545_CR15) 2015; 68 545_CR3 H Anai (545_CR10) 2000 V Weispfenning (545_CR11) 2000 LE Heindel (545_CR22) 1971; 18 545_CR2 A Khovanskii (545_CR1) 1980; 22 M Achatz (545_CR12) 2008 D Perrucci (545_CR21) 2011; 412 N Vorobjov (545_CR8) 1992; 13 J von zur Gathen (545_CR19) 2003 D Richardson (545_CR7) 1991 J van der Hoeven (545_CR17) 2006; 41 A Tarski (545_CR4) 1951 545_CR23 ML Barbagallo (545_CR16) 2016; 452 M-F Roy (545_CR13) 1994; 4 M Waldschmidt (545_CR18) 1978; 25 545_CR20 A Maignan (545_CR9) 1998 |
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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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