@misc{TsarevWolf, author = {Tsarev, Sergey and Wolf, Thomas}, title = {Hyperdeterminants as integrable discrete systems}, issn = {1438-0064}, doi = {10.1088/1751-8113/42/45/454023}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11292}, number = {09-17}, abstract = {We give the basic definitions and some theoretical results about hyperdeterminants, introduced by A.~Cayley in 1845. We prove integrability (understood as \$4d\$-consistency) of a nonlinear difference equation defined by the \$2 \times 2 \times 2\$ - hyperdeterminant. This result gives rise to the following hypothesis: the difference equations defined by hyperdeterminants of any size are integrable. We show that this hypothesis already fails in the case of the \$2\times 2\times 2\times 2\$ - hyperdeterminant.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {On solving large systems of polynomial equations appearing in Discrete Differential Geometry}, issn = {1438-0064}, doi = {10.1134/S0361768808020047}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10646}, number = {08-10}, abstract = {The paper describes a method for solution of very large overdetermined algebraic polynomial systems on an example that appears from a classification of all integrable 3-dimensional scalar discrete quasilinear equations \$Q_3=0\$ on an elementary cubic cell of the lattice \${\mathbb Z}^3\$. The overdetermined polynomial algebraic system that has to be solved is far too large to be formulated. A probing' technique which replaces independent variables by random integers or zero allows to formulate subsets of this system. An automatic alteration of equation formulating steps and equation solving steps leads to an iteration process that solves the computational problem.}, language = {en} } @misc{TsarevWolf, author = {Tsarev, Sergey and Wolf, Thomas}, title = {Classification of 3-dimensional integrable scalar discrete equations}, issn = {1438-0064}, doi = {10.1007/s11005-008-0230-2}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10667}, number = {08-13}, abstract = {We classify all integrable 3-dimensional scalar discrete affine linear equations \$Q_3=0\$ on an elementary cubic cell of the lattice \${\mathbb Z}^3\$. An equation \$Q_3=0\$ \%of such form is called integrable if it may be consistently imposed on all \$3\$-dimensional elementary faces of the lattice \${\mathbb Z}^4\$. Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only ontrivial(non-linearizable) integrable equation from this class is the well-known dBKP-system.}, language = {en} } @misc{WolfNeun, author = {Wolf, Thomas and Neun, Winfried}, title = {About a Computer Algebra based online Mathtest}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10688}, number = {08-14}, abstract = {The article describes the online mathematics test {\tt http://lie.math.brocku.ca/mathtest}, its typical applications and experiences gathered.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {The Parametric Solution of Underdetermined linear ODEs}, issn = {1438-0064}, doi = {10.1134/S0361768811020113}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10693}, number = {08-15}, abstract = {The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are arbitrary analytic functions in the independent variable. A second conceptual aim is to present an algorithm that is in some sense dual to the fundamental Euclids algorithm, and thus an alternative to the special case of a Gr\"{o}bner basis algorithm as it is used for solving linear ODE-systems. In the paper Euclids algorithm and the new dual version' are compared and their complementary strengths are analysed on the task of solving underdetermined ODEs. An implementation of the described algorithm is interactively accessible at http://lie.math.brocku.ca/crack/uode.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {Merging solutions of non-linear algebraic systems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11769}, number = {10-13}, abstract = {In solving large polynomial algebraic systems that are too big for standard Gr{\"o}bner basis techniques one way to make progress is to introduce case distinctions. This divide and conquer technique can be beneficial if the algorithms and computer programs know how to take advantage of inequalities. A further hurdle is the form of the resulting general solutions which often have unnecessarily many branches. In this paper we discuss a procedure to merge solutions by dropping inequalities which are associated with them and, if necessary, by re-parametrizing solutions. In the appendix the usefulness of the procedure is demonstrated in the classification of quadratic Hamiltonians with a Lie-Poisson bracket \$e(3)\$. This application required the solution of algebraic systems with over 200 unknowns, 450 equations and between 5000 and 9000 terms.}, language = {en} }