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Question: doubts about Groebner Basis over Galois (finite) field

Question:doubts about Groebner Basis over Galois (finite) field

gepo 541
February 25 2010
0
Hi, all, I have known how to compute Groebner basis over galois field. For example, the following example is to compute Groebner basis over GF(2^3): alias(alpha=RootOf(Z^3+Z+1)); F := {alpha*x*y+alpha^2+1, alpha^2*x^2-1}; G := Groebner[Basis](F, plex(x,y), characteristic=2); But now I have a doubt. If I give one more "alias", is the Groebner basis still computed over GF(2^3)? For example: alias(alpha=RootOf(Z^3+Z+1)); alias(beta=RootOf(u^2+u+1)); F := {alpha*x*y+alpha^2+1, alpha^2*x^2-1,alpha^2*beta+x*y}; G := Groebner[Basis](F, plex(x,y), characteristic=2); Is this Groebner basis computed over GF(2^3) or GF(2^2)? Another question, I heard that Maple's Groebner basis code did not use any special representations for elements of finite fields which made Maple slower for large problems than other systems, such as Magma. Is this true? How about Sage? Is Sage faster than Magma for running large problems? Thanks Gepo
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