A promenade through correct test sequences I: Degree of constructible sets, Bézout's Inequality and density
The notion of correct test sequence was introduced in [27]. It has been widely used to design probabilistic algorithms for Polynomial Identity Testing (PIT). In this manuscript we study foundations and generalizations of this notion. We show that correct test sequences are almost omnipresent in the...
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| Veröffentlicht in: | Journal of Complexity Jg. 68; S. 101588 |
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| Abstract | The notion of correct test sequence was introduced in [27]. It has been widely used to design probabilistic algorithms for Polynomial Identity Testing (PIT). In this manuscript we study foundations and generalizations of this notion. We show that correct test sequences are almost omnipresent in the mathematical literature: As hitting sets in PIT, in Function Identity Testing, as norming sets ([1]) or in Reproducing Kernel Hilbert Spaces context. We generalize the main statement of [27] proving that short correct test sequences for lists of polynomials are densely distributed in any constructible set of accurate co–dimension and degree. In order to prove this result we introduce and develop the theory of degree of constructible sets, generalizing [25] and proving two Bezout's Inequalities for two different notions of degree. We exhibit the strength and limitations of correct test sequences with a randomized efficient algorithm for the “Suite Sécante” Problem. Our algorithm generalizes [27], admitting a bigger class of sampling sets, also proving that PIT is in RPK. We reformulate, prove and generalize two results of the Polynomial Method: Dvir's exponential lower bounds for Kakeya sets and Alon's Combinatorial Nullstellensatz. |
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| AbstractList | The notion of correct test sequence was introduced in [27]. It has been widely used to design probabilistic algorithms for Polynomial Identity Testing (PIT). In this manuscript we study foundations and generalizations of this notion. We show that correct test sequences are almost omnipresent in the mathematical literature: As hitting sets in PIT, in Function Identity Testing, as norming sets ([1]) or in Reproducing Kernel Hilbert Spaces context. We generalize the main statement of [27] proving that short correct test sequences for lists of polynomials are densely distributed in any constructible set of accurate co–dimension and degree. In order to prove this result we introduce and develop the theory of degree of constructible sets, generalizing [25] and proving two Bezout's Inequalities for two different notions of degree. We exhibit the strength and limitations of correct test sequences with a randomized efficient algorithm for the “Suite Sécante” Problem. Our algorithm generalizes [27], admitting a bigger class of sampling sets, also proving that PIT is in RPK. We reformulate, prove and generalize two results of the Polynomial Method: Dvir's exponential lower bounds for Kakeya sets and Alon's Combinatorial Nullstellensatz. |
| ArticleNumber | 101588 |
| Author | Sebastián, Daniel Pardo, Luis M. |
| Author_xml | – sequence: 1 givenname: Luis M. surname: Pardo fullname: Pardo, Luis M. email: luis.m.pardo@gmail.com, luis.pardo@unican.es – sequence: 2 givenname: Daniel surname: Sebastián fullname: Sebastián, Daniel email: daniel.sebastian@unican.es |
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| Cites_doi | 10.1016/0304-3975(80)90019-5 10.1007/s10208-005-0176-2 10.1007/s10208-002-0065-7 10.1007/BF02579205 10.1090/S0273-0979-1989-15750-9 10.1016/j.jco.2021.101574 10.1016/j.tcs.2004.01.009 10.1016/S0304-3975(98)00261-8 10.1007/s00222-004-0434-8 10.1090/S0894-0347-08-00630-9 10.1007/s00037-013-0073-9 10.1017/S0963548398003411 10.1090/S0894-0347-08-00607-3 10.1006/aama.1998.0633 10.1016/S0022-4049(97)00015-7 10.2307/2372655 10.1145/322217.322225 10.1070/IM1969v003n05ABEH000834 10.1007/s10208-005-0211-0 10.1016/j.jco.2013.04.005 10.1137/08073408X 10.1006/aima.1996.1609 10.1016/0020-0190(78)90067-4 10.1016/S0764-4442(97)83558-6 10.1016/0304-3975(85)90150-1 10.1090/S0002-9904-1949-09219-4 10.1090/S0002-9947-1948-0026239-9 10.1017/CBO9780511618796 10.1090/S0002-9939-1954-0066627-6 10.4171/emss/1 10.1090/S0002-9947-1950-0051437-7 10.1016/0304-3975(83)90002-6 10.1016/j.jco.2019.04.001 10.1090/S0894-0347-1988-0944576-7 |
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| Keywords | Probability and existence Degree of constructible and locally closed affine sets Bounded error probability polynomial time algorithms Bézout's inequality Correct test sequences Equality test of lists of functions |
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| SubjectTerms | Bounded error probability polynomial time algorithms Bézout's inequality Correct test sequences Degree of constructible and locally closed affine sets Equality test of lists of functions Probability and existence |
| Title | A promenade through correct test sequences I: Degree of constructible sets, Bézout's Inequality and density |
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