Monic polynomial

In algebra, a monic polynomial is a polynomial

$c_nx^n+c_{n-1}x^{n-1}+\cdots+c_2x^2+c_1x+c_0$

in which the leading coefficient c_n is equal to 1.

If a polynomial has only one variable ( univariate polynomial ) , then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). A univariate polynomial in x of degree n then takes the general form displayed above, where

c_n ≠ 0, c_n−1, ..., c₂, c₁ and c₀

are constants, the coefficients of the polynomial.

Here the term c_nxⁿ is called the leading term, and its coefficient c_n the leading coefficient; if the leading coefficient is 1, the univariate polynomial is called monic.

Examples

• Monic quadratic polynomials

Properties

Multiplicatively closed

The set of all monic polynomials (over a given (unitary) ring A and for a given variable x) is closed under multiplication, since the product of the leading terms of two monic polynomials is the leading term of their product. Thus, the monic polynomials form a multiplicative semigroup of the polynomial ring A[x]. Actually, since the constant polynomial 1 is monic, this semigroup is even a monoid.

Polynomial equation solutions

In other respects, the properties of monic polynomials and of their corresponding monic polynomial equations depend crucially on the coefficient ring A. If A is a field, then every non-zero polynomial p has exactly one associated monic polynomial q; actually, q is p divided with its leading coefficient. In this manner, then, any non-trivial polynomial equation p(x) = 0 may be replaced by an equivalent monic equation q(x) = 0. E.g., the general real second degree equation

ax² + bx + c = 0 (where $a \neq 0$)

may be replaced by

x² + px + q = 0,

by putting  p = b/a  and  q = c/a. Thus, the equation 2x² + 3x + 1 = 0 is equivalent to the monic equation x² + 1.5x + 0.5 = 0.

Integrality

On the other hand, if the coefficient ring is not a field, there are more essential differences. E.g., a monic polynomial equation integer coefficients cannot have other rational solutions than integer solutions. Thus, the equation

2x² + 3x + 1 = 0

possibly might have some rational root, which is not an integer, (and incidently it does have inter alia the root -1/2); while the equations

x² + 5x + 6 = 0

and

x² + 7x + 8 = 0

only may have integer solutions or irrational solutions.

The solutions to monic polynomial equations over an integral domain are important in the theory for integral closures, and hence for algebraic number theory. In general, assume that A is an integral domain, and also a subring of the integral domain B. Consider the subset C of B, consisting of those B elements, which satisfy monic polynomial equations over A:

$C := \{b \in B : \exists\, p(x) \in A[x]\,, \hbox{ which is monic and such that } p(b) = 0\}\,.$

The set C contains A, since any a ∈ A satisfies the equation x − a = 0. Moreover, it is possible to prove that C is closed under addition and multiplication. Thus, C is a subring of B. The ring C is called the integral closure of A in B; or just the integral closure of A, if B is the fraction field of A; and the elements of C are said to be integral over A. If here A = Z (the ring of integers) and B = C (the field of complex numbers), then C is the ring of algebraic integers.

Ordinarily, the term monic is not employed for polynomials of several variables. However, a polynomial in several variables may be regarded as a polynominal 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

p(x,y) = 2xy² + x² − y² + 3x + 5y − 8

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

$p(x,y) = 1\cdot x^2 + (2y^2+3) \cdot x + (-y^2+5y-8)$;

but p(x,y) is not monic as an element in R[x][y], since then the highest degree coefficient (i.e., the y² coefficient) then 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 this alternative definition share some properties with the "ordinary" (univariate) monic polynomials. Notably, the product of monic polynomials again is monic.