Notes On Grobner Bases

B Mishra

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Thus h = {f-g)-{ai- a^Yfo = {f- aitfo) - {g - a^tfo) and we may choose /' = aitfo, g' = oj^/o- Q-E. D.
Let be the relation obtained as the union of > and its reverse: / g or g — > / holds.
Let *-^ denote the reflexive transitive closure of 0. The result is trivial for k = 0. Otherwise, by induction, 9\- 9k^ [F) and it is seen directly from the definition that go - Qi ^ [F). Thusgo-gte {F).
(=>) If f - g e {F) then we can express f - g a£ E. '^i oaf, where each Q, is a non-zero monomial and t...he fi are members of F. The /, 's are not necessarily distinct here. If m = the result is trivial. If m > 1 then we can write g' = g + amfm and f - g' = HZV "•/«• By induction hypothesis, / *^ g'.

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