Definitions for leading monomials, coefficients, and terms

Description

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The current terminology used in Groebner is that of Ideals, Varieties and Algorithms by David Cox, John Little, and Donal O'Shea, Springer-Verlag, (1992). In releases of Maple prior to Maple 10, Groebner used a different, conflicting convention. In particular, what is now called a leading monomial used to be called a leading term, and vice versa.

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Note: the old commands Groebner[leadmon] and Groebner[leadterm] still respect the old convention, however they are now deprecated. You should replace them with Groebner[LeadingTerm] or Groebner[LeadingMonomial] respectively.

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The current convention is as follows:

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A monomial is a product of indeterminates from a fixed set X, possibly with repetitions. The coefficient of a monomial is always one.

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A term of a polynomial (with respect to X) is the product of a monomial in X and a coefficient whose degree in X is zero. This coefficient may include other indeterminates not in X. For example, the coefficients may be rational functions in other variables.

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The "leading term" of a polynomial with respect to a monomial order is the term whose monomial is greatest with respect to the order and whose coefficient is non-zero. The coefficient and monomial of this term are called the "leading coefficient" and "leading monomial" of the polynomial, respectively.

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Note that the LeadingTerm command does not actually output terms, but rather the sequence (leading coefficient, leading monomial). This may be changed in a future release of Maple.

With respect to the definitions above, we will compute the leading coefficient, leading monomial, and leading term of the polynomial f with respect to lexicographic order with x > y > z > w > t.

In releases of Maple prior to Maple 10, Groebner[leadmon] computed what is now returned by LeadingTerm and Groebner[leadterm] computed what is now returned by LeadingMonomial.

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