%0 Report %T Sub-groups of Zn, standard basis, and linear diophantine systems %+ Algebraic Formal Systems for Industry and Research (SAFIR) %A Pottier, Loïc %N RR-1510 %P 11 %I INRIA %8 1991 %D 1991 %K Non disponible sur le PDF %Z Computer Science [cs]/Other [cs.OH]Reports %X We caracterize a (affine) sub-group of Zn by a polynomial ideal and standard (Gröbner) basis. We show how to use these standard basis to solve directly many algorithmic problems on (affine) sub-groups and their non-negative parts : * the membership problem * the triviality problem * find the smallest non zero vector * find the minimal non-negative elements * solve linear diohantine systems (Ax = 0, x>=0), (Ax = b, x>=0), Ax <= b on integers, and find smallest solutions Finally we conjecture other properties of these standard basis %G English %2 https://inria.hal.science/inria-00075052/document %2 https://inria.hal.science/inria-00075052/file/RR-1510.pdf %L inria-00075052 %U https://inria.hal.science/inria-00075052 %~ INRIA %~ INRIA-SOPHIA %~ INRIA-RRRT %~ INRIASO %~ INRIA_TEST %~ TESTALAIN1 %~ INRIA2 %~ LARA