Degree of a polynomial

The degree of a polynomial in one variable is the greatest exponent of mathematics in its standard representation as a sum of monomials. For example, the degree of the polynomial is equal to 5, that is the exponent of the monomial. When polynomials of several variables of the degrees of a monomial than the sum of the exponent of the variable potencies contained and the degree of a polynomial is defined (also referred to as total degree ) as the maximum of the degree of the monomers constituting the polynomial. Thus, for example, the monomial, and thus the degree of the polynomial 6

Definition

Be a commutative ring, a natural number and the polynomial ring in the variables. is

A monomial with, so the degree of is defined as

Now let

A polynomial, and monomials. Then, the degree or level of total is defined as

There are different conventions for the definition of the degree of. In algebra, it is customary to set. In contrast, the definition is in the areas of mathematics that deal with solving algebraic problems with the help of computers, often preferred.

Note: Since monomers consist of only a finite number of factors that can be the definition of the degree of a monomial and hence the definition of the degree of a polynomial directly extended to polynomial rings in any number of variables.

Properties

Be polynomials over. Then we have

• And
• .

In the event you can even see.

Is an integral domain, so even applies

For everyone.

Examples

Consider polynomials in (see integers ). It is

• ,
• ,
• And
• .