Albanian Journal of Mathematics (ISNN: 1930-1235), Vol 4, No 4 (2010)

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Conversion between Hermite and Popov normal forms using an FGLM-like approach

Johannes Middeke

Abstract


We are working with matrices over a ring K[\partial;\sigma,\theta] ofOre polynomials over a skew field K. Extending a result of Kojima etal. for usual polynomials it is shown that in this setting the Hermiteand Popov normal forms correspond to Gröbner bases with respect tocertain orders. The FGLM algorithm is adapted to this setting and usedfor converting Popov forms into Hermite forms and vice versa. Theapproach works for arbitrary, ie, not necessarily square matriceswhere we establish termination criteria to deal with infinitelydimensional factor spaces.

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