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The first (greatest) term of a polynomial p for this ordering and the corresponding monomial and coefficient are respectively called the leading term, leading monomial and leading coefficient and denoted, in this article, lt(p), lm(p) and lc(p). Most polynomial operations related to Gröbner bases involve the leading terms.
The leading term of a polynomial is thus the term of the largest monomial (for the chosen monomial ordering). Concretely, let R be any ring of polynomials. Then the set M of the (monic) monomials in R is a basis of R , considered as a vector space over the field of the coefficients.
The leading term of a nonzero polynomial [,, …,], denoted by () is the monomial term of maximal order in and the leading term of = is . The ideal of leading terms , denoted by L T ( I ) {\displaystyle LT(I)} , is the ideal generated by the leading terms of every element in the ideal, that is, L T ( I ) = ( L T ( f ) ∣ f ∈ I ...
The names for the degrees may be applied to the polynomial or to its terms. For example, the term 2x in x 2 + 2x + 1 is a linear term in a quadratic polynomial. The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero.
For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...
A crude version of this algorithm to find a basis for an ideal I of a polynomial ring R proceeds as follows: Input A set of polynomials F that generates I Output A Gröbner basis G for I. G := F; For every f i, f j in G, denote by g i the leading term of f i with respect to the given monomial ordering, and by a ij the least common multiple of g ...
Here are some examples. Every polynomial is associated to a unique monic polynomial. In particular, the unique factorization property of polynomials can be stated as: Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials.
The leading term of e λ t (X 1, ..., X n) is X λ. Proof. The leading term of the product is the product of the leading terms of each factor (this is true whenever one uses a monomial order, like the lexicographic order used here), and the leading term of the factor e i (X 1, ..., X n) is clearly X 1 X 2 ···X i.