Definitions for leading monomials, coefficients, and terms
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.
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.
The current convention is as follows:
A monomial is a product of indeterminates from a fixed set X, possibly with repetitions. The coefficient of a monomial is always one.
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.
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.
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.
f ≔ 5⁢x3⁢y+x2⁢w2⁢t+5⁢x3⁢y⁢z⁢t−2⁢x⁢z⁢w3⁢t+3⁢y2⁢w3⁢t
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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