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Leading Term


The leading term of a nonzero polynomial, relative to a specified monomial order, is its leading monomial together with its coefficient. Thus

 LT(f)=LC(f)LM(f),

where LM(f) is the leading monomial and LC(f) is the leading coefficient. For example, for f=3x^2y+2y^3 over Q with lexicographic order x>y, the leading term is 3x^2y, while the leading monomial is x^2y. For a monic polynomial, the leading term and leading monomial coincide.

The monomial order determines the word "leading," which need not mean highest total degree. For example, x is the leading term of x+y^(10) in lexicographic order with x>y. In the univariate polynomial case with the usual order by degree, the leading term of a_nx^n+...+a_0 is a_nx^n when a_n!=0.


See also

Leading Coefficient, Leading Monomial, Monomial Order, Polynomial Term, S-Polynomial

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References

Buchberger, B. "Gröbner Bases and Systems Theory." Multidimensional Systems Signal Process. 12, 223-251, 2001b. https://doi.org/10.1023/A:1011949421611.Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer-Verlag, 1997.

Cite this as:

Weisstein, Eric W. "Leading Term." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LeadingTerm.html

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