Algorithms and Implementations

Notes:

• If you would like to port some of this code to another Computer Algeba System, I would be happy to assist any way I can.
• E-mail me if you find broken links, or for related work, potential improvements, test files, etc.
• Old version of this page.

Algorithms with former students:

• Note: The algorithms listed below with Imamoglu, Kunwar, Yuan, and Debeerst, have now been combined into a single package that will be part of Maple 2021.
• Jonas Szutkoski (PhD 2017). Implementations for computing subfields, and computing rational function decomposition (now part of Magma).
• Erdal Imamoglu (PhD 2017). Implementation for finding hypergeometric solutions of second order linear differential equations, in Maple 2020.
• Wen Xu (PhD 2017). Implementation: two solvers and tools for A-hypergeometric functions.
• Vijay Kunwar (PhD 2014). Implementations for 2F1 type solutions for deg(pullback) <= 3, and for equations with at most 5 true singularities, see FiveSing database.
• Tingting Fang (PhD 2012) (ISSAC'2011, thesis, slides). Find 2-descent for a differential equation, e-mail for implementation.
• Quan Yuan (PhD 2012) and Ruben Debeerst (a former visiting undergraduate student). Implementation for computing Airy/Bessel/Kummer/Whittaker type solutions.
• Vivek Pal (at the time an undergraduate at FSU). Implementation for Isomorphisms of Algebraic Number Fields.
• Yongjae Cha (PhD 2010). Implementation for closed form solutions of Linear Difference Equations. (2012): implementation of algorithm Hom
• Giles Levy (PhD 2009). Implementation to find relations to OEIS sequences, and fastest algorithm for Liouvillian Solutions (for 2nd order Difference Equations).
• Andy Novocin (PhD 2008). Analyzed the complexity of factoring in Q[x] which led to faster lattice reduction.

• Fastest algorithm to factor in Q[x] is in Maple, Magma, NTL, FLINT, PARI/GP, and Sage and on French wikipedia.
• Bivariate polynomial factorization, in Magma and in Singular.
• Modular GCD (implementated in Maple and Magma by Michael Monagan, cited in Microsoft patent).
• Gradual sub-lattice reduction (with Andy Novocin)
• Isomorphisms between number fields (with Vivek Pal)
• Subfields and decomposing rational functions (with Jonas Szutkoski)

• Puiseux Expansion and Integral Basis, paper and implementation (1994).
• Parametrization of Rational Algebraic Curves, papers and implementations (1994,1997).
• Weierstrass Normal Form for elliptic (1995) and hyper-elliptic (1999) curves, papers and implementations.
• Overview of algcurves in 1996 (try plot_knot!).
• Joint with Bernard Deconinck: Riemann Matrices (2001) and Riemann Theta Functions (2004).
• FindDessins, Maple 2019 help page, source code, is an algorithm used to classify Belyi maps for the FiveSing database (2016).

• Algorithms (2013) such as ComputeBelyi for the Heun database with R. Vidunas.
• Integral basis in Characteristic p (with Mike Stillman) (nearly complete implementation available by e-mail)
• Finding a point on a conic over a rational function field (2006). Similar topic: ConicIsomorphism.
• Abel's problem (2000) (minpoly of exp(int(a,x)) where a is an algebraic function).

• diffop package which includes
  • DFactor. Cited by Mathematica. Special code for order 4.
  • eigenring and DFactorLCLM
  • formal solutions and generalized exponents
  • integrate_sols
  • expsols. Improvements (not implemented) in CH'2004.
  • Hom (there is no paper for this in the differential case, only for the difference case)
  • other tools (adjoint, mult, GCRD, LCLM, rightdivision, symmetric product, symmetric/exterior power).
• Liouvillian solutions for order 2 (paper) and imprimitive case for order 3.
• Detecting a symmetric product or symmetric power of second order operators and other patterns.
• FindODE, Maple 2019 help page (last example is solved due to symmetric product program in the previous item), source code.

• ReduceOrder, detects if a 3rd order equation is gauge-equivalent to a symmetric square of a 2nd order equation.
• Desingularization
• Compute invariants of the differential Galois group.
• Solving 2nd order equations in closed form, see "Algorithms with former students" above.

• Hypergeometric solutions. Cited by Mathematica. Improvements (not implemented) in CH'2006.
• Desingularization
• Rational definite summation
• Hom, Liouvillian and other closed-form solutions, see "Algorithms with former students" above.
• Factoring Recurrence Operators (will be part of Maple 2021).

• Modular curve X1(N): Gonality (with Derickx), and Low Degree Places.
• Belyi and near Belyi maps with four resp. five exceptional points, with Vidunas resp. Kunwar.
• Optimized equations for X1(N) (with Sutherland) and for X_H(l) (to compute Galois representations, with Derickx and Zeng).