Refine
Year of publication
Document Type
- ZIB-Report (43)
- Article (12)
Language
- English (55)
Is part of the Bibliography
- no (55)
Keywords
- computer algebra (6)
- conservation laws (3)
- discrete equations (3)
- integrable systems (3)
- large polynomial systems (3)
- Computer Go (2)
- Computer algebra (2)
- Crack (2)
- Integrability (2)
- REDUCE (2)
Institute
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide.
The algorithmic aspect of our work provides a method to compute rational solutions of single polynomial equations that are typically large with 10^2 ... 10^5 terms and that are heavily underdetermined.
It was possible to obtain this functionality by adding a number of new modules for a new type of splitting of equations to the existing package CRACK that is normally used to solve polynomial algebraic and differential systems of equations.
Recently, Holm and Ivanov, proposed and studied a class of multi-component generalizations of the Camassa–Holm equations [D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys A: Math. Theor. 43 (2010) 492001 (20pp)]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete set of first-order integrating factors for the systems in Cauchy–Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2).
Recently, Holm and Ivanov, proposed and studied a class of multi-component
generalisations of the Camassa-Holm equations [D D Holm and R I Ivanov,
Multi-component generalizations of the CH equation: geometrical
aspects, peakons and numerical examples,
{\it J. Phys A: Math. Theor} {\bf 43}, 492001 (20pp), 2010]. We
consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and
CH(2,2), and report a class of integrating factors and its
corresponding conservation laws for these two systems. In particular,
we obtain
the complete sent of first-order integrating factors for the systems
in Cauchy-Kovalevskaya form and evaluate the corresponding sets of
conservation laws for CH(2,1) and CH(2,2).
The paper reports on a computer algebra program {\sc LSSS} (Linear Selective
Systems Solver) for solving linear algebraic systems with rational
coefficients. The program is especially efficient for very large sparse
systems that have a solution in which many variables take the value
zero. The program is applied to the symmetry investigation of a non-abelian
Laurent ODE introduced recently by M.\ Kontsevich. The computed symmetries
confirmed that a Lax pair found for this system earlier generates all first
integrals of degree at least up to 14.