Editing Monic polynomial (section)

== 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 one variable with coefficients being polynomials in the other variables. Being ''monic'' depends thus on the choice of one "main" variable. For example, the polynomial
:<math>p(x,y) = 2xy^2+x^2-y^2+3x+5y-8</math>
is monic, if considered as a polynomial in {{mvar|x}} with coefficients that are  polynomials in {{mvar|y}}:
:<math>p(x,y) = x^2 + (2y^2+3) \, x + (-y^2+5y-8);</math>
but it is not monic when considered as a polynomial in {{mvar|y}} with coefficients polynomial in {{mvar|x}}:
:<math>p(x,y)=(2x-1)\,y^2+5y +(x^2+3x-8).</math>

In the context of [[Gröbner bases]], a [[monomial order]] is generally fixed. In this case, a polynomial may be said to be monic, if it has 1 as its leading coefficient (for the monomial order).

For every definition, a product of monic polynomials is monic, and, if  the coefficients belong to a [[field (mathematics)|field]], every polynomial is [[associated element|associated]] to exactly one monic polynomial.