Let α := 2 cos(2π/7) and let τ be the golden ratio, they are solutions of α 3 + α 2 − 2α − 1 = 0,... more Let α := 2 cos(2π/7) and let τ be the golden ratio, they are solutions of α 3 + α 2 − 2α − 1 = 0, τ 2 − τ − 1 = 0. Let b := (6τ − 3)α 2 + (14τ − 8)α + 5τ − 3 and c := τ α 2 + 2τ α + 1. Let E b,c be the elliptic curve E b,c := y 2 + (1 − c)xy − by = x 3 − bx 2 (1) and P := (x = 0, y = 0).
A tool package for computing genus 0 Belyi functions is presented, including simplification routi... more A tool package for computing genus 0 Belyi functions is presented, including simplification routines, computation of moduli fields, decompositions, dessins d'enfant. The main algorithm for computing the Belyi functions themselves is based on implied transformations of the hypergeometric differential equation to Fuchsian equations, preferably with few singular points. This gives interesting differential relations between polynomial components of a Belyi function.
We classify all rational functions f : P 1 → P 1 whose branching pattern above 0, 1, ∞ satisfy a ... more We classify all rational functions f : P 1 → P 1 whose branching pattern above 0, 1, ∞ satisfy a certain regularity condition with precisely d = 5 exceptions. This work is motivated by solving second order linear differential equations, with d = 5 true singularities, in terms of hypergeometric functions. A similar problem was solved for d = 4 in [2]. * Supported by NSF grants 1319547 and 1618657. 1 We focus on the regular singular case because for irregular singular equations of order 2, a complete algorithm to find all {Airy, Bessel, Kummer, Whittaker}-type solutions was given in [16, 9]. The regular-singular assumption can be replaced by the assumption that y in Conjecture 1 has a non-zero radius of convergence. 2 This condition implies at least one logarithmic singularity. Because of the conjecture we focus on differential equations with at least one logarithmic singularity, however, the same table can also be used for more general "parametric" cases [2]. 3 Equations with a convergent integer power series solution (globally nilpotent differential equations). 4 Solutions related to entries of the diagram can be expressed in terms of those three entries. However, the other entries can still be relevant if we want solutions of minimal size, see Section 5.3.3 (decompositions) in [11] for more. Entry (∞, ∞, ∞) corresponds to writing solutions in terms of the elliptic integrals K and E.
Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the ... more Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the p-adic integers). Current implementations Hensel lift to determine f1,...,fr with a p-adic accuracy a that is guaranteed to be high enough to recover any potential factor of f in Z[x]. However, the problem is that this p-adic accuracy, a, is often much higher than what was actually necessary to recover all the factors g1,...,g k. This implies that current implementations often waste CPU time on Hensel Lifting. In practice it frequently happens that f has one large factor, say g1, and zero or more small factors, say g2,...,g k. Then, to recover g1,...,g k we do not need p a to be larger than twice the largest coefficient of g1. All we need is that p a is larger than twice the largest coefficient in g2,...,g k. This suffices to reconstruct g2,...,g k ∈ Z[x] from their modular images, after which the remaining factor g1 can be determined by a division in Z[x]. It is easy to give examples wh...
Transcendence in Algebra, Combinatorics, Geometry and Number Theory, 2021
This paper gives a quick way to construct all modular functions for the group Γ 0 (N) having only... more This paper gives a quick way to construct all modular functions for the group Γ 0 (N) having only a pole at τ = i∞. We assume that we are given two modular functions f, g for Γ 0 (N) with poles only at i∞ and coprime pole orders. As an application we obtain two new identities from which one can derive that p(11n + 6) ≡ 0 (mod 11), here p(n) is the usual partition function.
Let K = Q(t1,. .. , t k) and a, b, c ∈ K. We give a simple algorithm to find, if it exists, X, Y,... more Let K = Q(t1,. .. , t k) and a, b, c ∈ K. We give a simple algorithm to find, if it exists, X, Y, Z in K, not all zero, for which aX 2 + bY 2 + cZ 2 = 0.
Let a n denote the number of ways in which a chess rook can move from a corner cell to the opposi... more Let a n denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an n × n × n three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven discovery and proof of the fact that the generating series G(x) = n≥0 a n x n admits the following explicit expression in terms of a Gaussian hypergeometric function: G(x) = 1 + 6 • x 0 2 F 1 1/3 2/3 2 27w(2−3w) (1−4w) 3 (1 − 4w)(1 − 64w) dw.
Proceedings of the 1996 international symposium on Symbolic and algebraic computation - ISSAC '96, 1996
The topic of this paper is a fast method to compute the rational solutions of a certain different... more The topic of this paper is a fast method to compute the rational solutions of a certain differential equation that will be called the mixed differential equation. This can be applied to speed up the factorization of completely reducible linear differential operators with rational functions coefficients.
Proceedings of the 2001 international symposium on Symbolic and algebraic computation, 2001
A new algorithm is presented for factoring bivariate approximate polynomials over C[a:, y]. Given... more A new algorithm is presented for factoring bivariate approximate polynomials over C[a:, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the degree of the problem. This method has been implemented in Maple, and has been demonstrated to be efficient and numerically robust.
Differential equations with 2F1-type solutions are very common in Mathematics and they also occur... more Differential equations with 2F1-type solutions are very common in Mathematics and they also occur in several areas such as Physics and Combinatorics. We are interested in finding 2F1-type solutions of second order differential equations. Paper [1] gives an algorithm to find 2F1-type solutions of second order differential equations when the degree of the pullback function1 of f is three. In our algorithm, the pullback function f is of arbitrary degree (however, we restrict to a special case defined below).
Proceedings of the 2004 international symposium on Symbolic and algebraic computation, 2004
We consider the problem of finding closed form solutions of linear differential equations having ... more We consider the problem of finding closed form solutions of linear differential equations having coefficients which are elliptic functions. For second order equations we show how to solve such an ode in terms of doubly periodic functions of the second kind. The method depends on two procedures, the first using a second symmetric power of an ode along with a decision procedure for determining when such equations have elliptic function solutions while the second involves the computation of exponential solutions.
Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, 2013
This tutorial will explain the algorithm behind the currently fastest implementations for univari... more This tutorial will explain the algorithm behind the currently fastest implementations for univariate factorization over the rationals. The complexity will be analyzed; it turns out that modifications were needed in order to prove a polynomial time complexity while preserving the best practical performance. The complexity analysis leads to two results: (1) it shows that the practical performance on common inputs can be improved without harming the worst case performance, and (2) it leads to an improved complexity, not only for factoring, but for LLL reduction as well.
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation, 2008
For differential operators of order 2, this paper presents a new method that combines generalized... more For differential operators of order 2, this paper presents a new method that combines generalized exponents to find those solutions that can be represented in terms of Bessel functions.
Proceedings of the 1998 international symposium on Symbolic and algebraic computation, 1998
This paper presents a new and sharper bound for denominators of rational solutions of linear di e... more This paper presents a new and sharper bound for denominators of rational solutions of linear di erence and q-di erence equations. This can be used to compute rational solutions more e ciently.
Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94, 1994
In this paper I want to present a new method for computing parametrizations of algebraic curves. ... more In this paper I want to present a new method for computing parametrizations of algebraic curves. Basically this method is a direct application of integral basis computation. Examples show that this method is faster than older methods.
Proceedings of the 2004 international symposium on Symbolic and algebraic computation, 2004
Let L be an algebraic function field in k ≥ 0 parameters t1,. .. , t k. Let f1, f2 be non-zero po... more Let L be an algebraic function field in k ≥ 0 parameters t1,. .. , t k. Let f1, f2 be non-zero polynomials in L[x]. We give two algorithms for computing their gcd. The first, a modular GCD algorithm, is an extension of the modular GCD algorithm of Brown for Z[x1,. .. , xn] and Encarnacion for Q(α)[x] to function fields. The second, a fraction-free algorithm, is a modification of the Moreno Maza and Rioboo algorithm for computing gcds over triangular sets. The modification reduces coefficient growth in L to be linear. We give an empirical comparison of the two algorithms using implementations in Maple.
The software is an implementation of the algorithms in [1], [2], and [3]. The main algorithm from... more The software is an implementation of the algorithms in [1], [2], and [3]. The main algorithm from [3] is implemented with additional base equations beyond what appear in [3] and is incorporated into [4]. Common to each algorithm is a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gt-transformations).
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, 2010
In this paper we give a new algorithm to compute Liouvillian solutions of linear difference equat... more In this paper we give a new algorithm to compute Liouvillian solutions of linear difference equations. The first algorithm for this was given by Hendriks in 1998, and Hendriks and Singer in 1999. Several improvements have been published, including a paper by Cha and van Hoeij that reduces the combinatorial problem. But the number of combinations still depended exponentially on the number of singularities. For irreducible second order equations, we give a short and very efficient algorithm; the number of combinations is 1.
Proceedings of the 2009 international symposium on Symbolic and algebraic computation, 2009
In this paper we give a new algorithm to compute Liouvillian solutions of linear difference equat... more In this paper we give a new algorithm to compute Liouvillian solutions of linear difference equations. Compared to the prior algorithm by Hendriks and Singer, our main contribution consists of two theorems that significantly reduce the number of combinations that the algorithm will check.
Let α := 2 cos(2π/7) and let τ be the golden ratio, they are solutions of α 3 + α 2 − 2α − 1 = 0,... more Let α := 2 cos(2π/7) and let τ be the golden ratio, they are solutions of α 3 + α 2 − 2α − 1 = 0, τ 2 − τ − 1 = 0. Let b := (6τ − 3)α 2 + (14τ − 8)α + 5τ − 3 and c := τ α 2 + 2τ α + 1. Let E b,c be the elliptic curve E b,c := y 2 + (1 − c)xy − by = x 3 − bx 2 (1) and P := (x = 0, y = 0).
A tool package for computing genus 0 Belyi functions is presented, including simplification routi... more A tool package for computing genus 0 Belyi functions is presented, including simplification routines, computation of moduli fields, decompositions, dessins d'enfant. The main algorithm for computing the Belyi functions themselves is based on implied transformations of the hypergeometric differential equation to Fuchsian equations, preferably with few singular points. This gives interesting differential relations between polynomial components of a Belyi function.
We classify all rational functions f : P 1 → P 1 whose branching pattern above 0, 1, ∞ satisfy a ... more We classify all rational functions f : P 1 → P 1 whose branching pattern above 0, 1, ∞ satisfy a certain regularity condition with precisely d = 5 exceptions. This work is motivated by solving second order linear differential equations, with d = 5 true singularities, in terms of hypergeometric functions. A similar problem was solved for d = 4 in [2]. * Supported by NSF grants 1319547 and 1618657. 1 We focus on the regular singular case because for irregular singular equations of order 2, a complete algorithm to find all {Airy, Bessel, Kummer, Whittaker}-type solutions was given in [16, 9]. The regular-singular assumption can be replaced by the assumption that y in Conjecture 1 has a non-zero radius of convergence. 2 This condition implies at least one logarithmic singularity. Because of the conjecture we focus on differential equations with at least one logarithmic singularity, however, the same table can also be used for more general "parametric" cases [2]. 3 Equations with a convergent integer power series solution (globally nilpotent differential equations). 4 Solutions related to entries of the diagram can be expressed in terms of those three entries. However, the other entries can still be relevant if we want solutions of minimal size, see Section 5.3.3 (decompositions) in [11] for more. Entry (∞, ∞, ∞) corresponds to writing solutions in terms of the elliptic integrals K and E.
Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the ... more Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the p-adic integers). Current implementations Hensel lift to determine f1,...,fr with a p-adic accuracy a that is guaranteed to be high enough to recover any potential factor of f in Z[x]. However, the problem is that this p-adic accuracy, a, is often much higher than what was actually necessary to recover all the factors g1,...,g k. This implies that current implementations often waste CPU time on Hensel Lifting. In practice it frequently happens that f has one large factor, say g1, and zero or more small factors, say g2,...,g k. Then, to recover g1,...,g k we do not need p a to be larger than twice the largest coefficient of g1. All we need is that p a is larger than twice the largest coefficient in g2,...,g k. This suffices to reconstruct g2,...,g k ∈ Z[x] from their modular images, after which the remaining factor g1 can be determined by a division in Z[x]. It is easy to give examples wh...
Transcendence in Algebra, Combinatorics, Geometry and Number Theory, 2021
This paper gives a quick way to construct all modular functions for the group Γ 0 (N) having only... more This paper gives a quick way to construct all modular functions for the group Γ 0 (N) having only a pole at τ = i∞. We assume that we are given two modular functions f, g for Γ 0 (N) with poles only at i∞ and coprime pole orders. As an application we obtain two new identities from which one can derive that p(11n + 6) ≡ 0 (mod 11), here p(n) is the usual partition function.
Let K = Q(t1,. .. , t k) and a, b, c ∈ K. We give a simple algorithm to find, if it exists, X, Y,... more Let K = Q(t1,. .. , t k) and a, b, c ∈ K. We give a simple algorithm to find, if it exists, X, Y, Z in K, not all zero, for which aX 2 + bY 2 + cZ 2 = 0.
Let a n denote the number of ways in which a chess rook can move from a corner cell to the opposi... more Let a n denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an n × n × n three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven discovery and proof of the fact that the generating series G(x) = n≥0 a n x n admits the following explicit expression in terms of a Gaussian hypergeometric function: G(x) = 1 + 6 • x 0 2 F 1 1/3 2/3 2 27w(2−3w) (1−4w) 3 (1 − 4w)(1 − 64w) dw.
Proceedings of the 1996 international symposium on Symbolic and algebraic computation - ISSAC '96, 1996
The topic of this paper is a fast method to compute the rational solutions of a certain different... more The topic of this paper is a fast method to compute the rational solutions of a certain differential equation that will be called the mixed differential equation. This can be applied to speed up the factorization of completely reducible linear differential operators with rational functions coefficients.
Proceedings of the 2001 international symposium on Symbolic and algebraic computation, 2001
A new algorithm is presented for factoring bivariate approximate polynomials over C[a:, y]. Given... more A new algorithm is presented for factoring bivariate approximate polynomials over C[a:, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the degree of the problem. This method has been implemented in Maple, and has been demonstrated to be efficient and numerically robust.
Differential equations with 2F1-type solutions are very common in Mathematics and they also occur... more Differential equations with 2F1-type solutions are very common in Mathematics and they also occur in several areas such as Physics and Combinatorics. We are interested in finding 2F1-type solutions of second order differential equations. Paper [1] gives an algorithm to find 2F1-type solutions of second order differential equations when the degree of the pullback function1 of f is three. In our algorithm, the pullback function f is of arbitrary degree (however, we restrict to a special case defined below).
Proceedings of the 2004 international symposium on Symbolic and algebraic computation, 2004
We consider the problem of finding closed form solutions of linear differential equations having ... more We consider the problem of finding closed form solutions of linear differential equations having coefficients which are elliptic functions. For second order equations we show how to solve such an ode in terms of doubly periodic functions of the second kind. The method depends on two procedures, the first using a second symmetric power of an ode along with a decision procedure for determining when such equations have elliptic function solutions while the second involves the computation of exponential solutions.
Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, 2013
This tutorial will explain the algorithm behind the currently fastest implementations for univari... more This tutorial will explain the algorithm behind the currently fastest implementations for univariate factorization over the rationals. The complexity will be analyzed; it turns out that modifications were needed in order to prove a polynomial time complexity while preserving the best practical performance. The complexity analysis leads to two results: (1) it shows that the practical performance on common inputs can be improved without harming the worst case performance, and (2) it leads to an improved complexity, not only for factoring, but for LLL reduction as well.
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation, 2008
For differential operators of order 2, this paper presents a new method that combines generalized... more For differential operators of order 2, this paper presents a new method that combines generalized exponents to find those solutions that can be represented in terms of Bessel functions.
Proceedings of the 1998 international symposium on Symbolic and algebraic computation, 1998
This paper presents a new and sharper bound for denominators of rational solutions of linear di e... more This paper presents a new and sharper bound for denominators of rational solutions of linear di erence and q-di erence equations. This can be used to compute rational solutions more e ciently.
Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94, 1994
In this paper I want to present a new method for computing parametrizations of algebraic curves. ... more In this paper I want to present a new method for computing parametrizations of algebraic curves. Basically this method is a direct application of integral basis computation. Examples show that this method is faster than older methods.
Proceedings of the 2004 international symposium on Symbolic and algebraic computation, 2004
Let L be an algebraic function field in k ≥ 0 parameters t1,. .. , t k. Let f1, f2 be non-zero po... more Let L be an algebraic function field in k ≥ 0 parameters t1,. .. , t k. Let f1, f2 be non-zero polynomials in L[x]. We give two algorithms for computing their gcd. The first, a modular GCD algorithm, is an extension of the modular GCD algorithm of Brown for Z[x1,. .. , xn] and Encarnacion for Q(α)[x] to function fields. The second, a fraction-free algorithm, is a modification of the Moreno Maza and Rioboo algorithm for computing gcds over triangular sets. The modification reduces coefficient growth in L to be linear. We give an empirical comparison of the two algorithms using implementations in Maple.
The software is an implementation of the algorithms in [1], [2], and [3]. The main algorithm from... more The software is an implementation of the algorithms in [1], [2], and [3]. The main algorithm from [3] is implemented with additional base equations beyond what appear in [3] and is incorporated into [4]. Common to each algorithm is a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gt-transformations).
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, 2010
In this paper we give a new algorithm to compute Liouvillian solutions of linear difference equat... more In this paper we give a new algorithm to compute Liouvillian solutions of linear difference equations. The first algorithm for this was given by Hendriks in 1998, and Hendriks and Singer in 1999. Several improvements have been published, including a paper by Cha and van Hoeij that reduces the combinatorial problem. But the number of combinations still depended exponentially on the number of singularities. For irreducible second order equations, we give a short and very efficient algorithm; the number of combinations is 1.
Proceedings of the 2009 international symposium on Symbolic and algebraic computation, 2009
In this paper we give a new algorithm to compute Liouvillian solutions of linear difference equat... more In this paper we give a new algorithm to compute Liouvillian solutions of linear difference equations. Compared to the prior algorithm by Hendriks and Singer, our main contribution consists of two theorems that significantly reduce the number of combinations that the algorithm will check.
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Papers by Mark Hoeij