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Background on Ideals and Gröbner Bases

The Gröbner Basis Algorithm requires that we have imposed an ordering (a ``term order'') on the monic monomials in the ring of polynomials in several variables tex2html_wrap_inline3944 over a field K as in §. Once a term order is defined, one can define the notion of a leading term, leading monomial and leading coefficient of a nonzero polynomial p. If p is a polynomial, tex2html_wrap_inline6428 and tex2html_wrap_inline6430 is a monic monomial, then tex2html_wrap_inline6432 is the leading term of p if tex2html_wrap_inline6432 appears as a term of p and for every term c M of p (with tex2html_wrap_inline6444 and M a monic monomial), tex2html_wrap_inline6430 is greater than or equal to M in the term order. In this case, tex2html_wrap_inline6452 is called the leading coefficient of p and tex2html_wrap_inline6430 is called the leading monomial of p. For a nonzero polynomial f, let lt(f) denote the leading term of f (with respect to the given ordering), lc(f) denote the leading coefficient of f and lm(f) denote the leading monomial of f. Note that lt(f)=lc(f) lm(f).

Wed Jul 3 10:27:42 PDT 1996