Computing Characteristic Polynomials of p-Curvatures in Average Polynomial Time

Pagès, Raphaël

doi:10.1145/3452143.3465524

Cited by 5 publications

(5 citation statements)

References 18 publications

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“…In [3], -characteristic polynomials of recurrence operators (and differential operators) over F [ ] are studied and an algorithm for computing them is given. An algorithm for computing -characteristic polynomials of operators in Z[ ] [ ] for a number of is presented in [10], based on the algorithm from [3]. We will give more information about these algorithms in subsection 2.4…”

Section: P-characteristic Polynomialmentioning

confidence: 99%

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Desingularization and p-Curvature of Recurrence Operators

Zhou¹,

Hoeij²

2022

Preprint

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Linear recurrence operators in characteristic are classified by their -curvature. For a recurrence operator , denote by ( ) the characteristic polynomial of its -curvature. We can obtain information about the factorization of by factoring ( ). The main theorem of this paper gives an unexpected relation between ( ) and the true singularities of . An application is to speed up a fast algorithm for computing ( ) by desingularizing first. Another contribution of this paper is faster desingularization. CCS CONCEPTS• Computing methodologies → Algebraic algorithms.

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Section: P-characteristic Polynomialmentioning

confidence: 99%

“…For an operator ∈ Q( )[ ], denote by ( ) the -characteristic polynomial of its reduction modulo . Pagès (2021) gives an algorithm for computing ( ) for a number of primes at the same time, if ∈ Z[ ] [ ] has a leading coefficient in Z ([10, Algorithm 3]). The algorithm is based on the BCS algorithm.…”

Section: Bcs Algorithm and Pagès' Algorithmmentioning

confidence: 99%

Desingularization and p-Curvature of Recurrence Operators

Zhou¹,

Hoeij²

2022

Preprint

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“…given. An algorithm for computing 𝑝-characteristic polynomials of operators in Z[𝑥] [𝜏] for a number of 𝑝 is presented in [10], based on the algorithm from [3]. We will give more information about these algorithms in subsection 2.4.…”

Section: P-characteristic Polynomialmentioning

confidence: 99%

Desingularization and p-Curvature of Recurrence Operators

Zhou

Hoeij

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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Linear recurrence operators in characteristic 𝑝 are classified by their 𝑝-curvature. For a recurrence operator 𝐿, denote by 𝜒 (𝐿) the characteristic polynomial of its 𝑝-curvature. We can obtain information about the factorization of 𝐿 by factoring 𝜒 (𝐿). The main theorem of this paper gives an unexpected relation between 𝜒 (𝐿) and the true singularities of 𝐿. An application is to speed up a fast algorithm for computing 𝜒 (𝐿) by desingularizing 𝐿 first. Another contribution of this paper is faster desingularization. CCS CONCEPTS• Computing methodologies → Algebraic algorithms.

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“…The amount of overdetermination of the linear system was already mentioned as a source of trust. Some further tests that may help to distinguish correct equations from noise have been proposed in [9]: a correct equation is likely to contain short integer coefficients while a wrong guess will typically contain long integers; a correct recurrence for an integer sequence must produce only integers when it is unrolled while a wrong guess will typically produce rational numbers; a correct differential equation for a generating function is likely to have nice singularities while a wrong guess will typically have awkward singularities; a correct differential equation for a generating function of an integer sequence with moderate growth must have nilpotent p-curvature [6,7,24] while a wrong guess will typically not have this property. All these tests are extremely strong and make guessing a very reliable tool in practice.…”

Section: Introductionmentioning

confidence: 99%