Monic Polynomial - Multivariate Polynomials

Multivariate Polynomials

Ordinarily, the term monic is not employed for polynomials of several variables. However, a polynomial in several variables may be regarded as a polynomial in only "the last" variable, but with coefficients being polynomials in the others. This may be done in several ways, depending on which one of the variables is chosen as "the last one". E.g., the real polynomial

is monic, considered as an element in R, i.e., as a univariate polynomial in the variable x, with coefficients which themselves are univariate polynomials in y:

;

but p(x,y) is not monic as an element in R, since then the highest degree coefficient (i.e., the y2 coefficient) is 2x − 1.

There is an alternative convention, which may be useful e.g. in Gröbner basis contexts: a polynomial is called monic, if its leading coefficient (as a multivariate polynomial) is 1. In other words, assume that p = p(x₁,...,x_n) is a non-zero polynomial in n variables, and that there is a given monomial order on the set of all ("monic") monomials in these variables, i.e., a total order of the free commutative monoid generated by x₁,...,x_n, with the unit as lowest element, and respecting multiplication. In that case, this order defines a highest non-vanishing term in p, and p may be called monic, if that term has coefficient one.

"Monic multivariate polynomials" according to either definition share some properties with the "ordinary" (univariate) monic polynomials. Notably, the product of monic polynomials again is monic.