Homogeneous polynomial

On this post we explain what homogeneous polynomials are. You will also see examples of homogeneous polynomials and the properties of this type of polynomial. Iin addition, you will find what the difference between homogeneous polynomials and heterogeneous polynomials is.

What is a homogeneous polynomial?

The definition of a homogeneous polynomial is as follows:

In mathematics, a homogeneous polynomial is polynomial in which all its terms are of the same degree.

An example of a homogeneous polynomial is:

 P(x,y,z)=x^3+5x^2y-4xyz

In this case, it is a homogeneous polynomial of degree 3, since all the monomials that are part of the polynomial are of third degree.

Examples of homogeneous polynomials

Once we have seen what a homogeneous polynomial means, let’s look at a few examples of homogeneous polynomials to understand better the concept:

  • Example of a homogeneous polynomial of degree 2:

P(x,y)=x^2+9xy-5y^2

  • Example of a homogeneous polynomial of degree 5:

P(x,y)=x^5+3x^2y^3-6x^4y+10xy^4

  • Example of a homogeneous polynomial of degree 7:

P(x,y,z)=x^3y^4+2x^5y^2+4x^2y^2z^3-x^2y^4z

  • Example of a homogeneous polynomial of degree 13:

P(a,b,c)=7a^6b^4c^3+2a^8b^3c^2+5a^4b^8c

Homogeneous polynomial and heterogeneous polynomial

Another polynomial very similar to the homogeneous polynomial is the heterogeneous polynomial, although there is a fundamental difference between them:

A heterogeneous polynomial is a polynomial which not all of its terms have the same degree.

Therefore, if only a monomial of the polynomial has a different degree than the rest of the terms, that polynomial will be heterogeneous.

For example, the following polynomial is heterogeneous:

P(x,y)=x^4+2x^3y+8x^2

Although two of the terms of the polynomial are of degree 4 (x4, 2x3y), it is a heterogeneous polynomial because it has another term whose degree is different (the term 8x2 is of second degree).

As you can see, homogeneous and heterogeneous polynomials are very similar to each other and are easily confused, so we must pay attention.

Properties of homogeneous polynomials

Homogeneous polynomials have the following characteristics

  • The number of different homogeneous monomials of degree M in a polynomial of N variables can be calculated using the following formula:

 \cfrac{(M+N-1)!}{M!(N-1)!}

The “!” sign may seem strange to you. Well, you should know that it is used to indicate a special mathematical operation, which is the factorial of a number.

  • The expression of the Taylor series that corresponds to a homogeneous polynomial expanded to the point x is as follows:

P(x+y)= \sum_{j=0}^n {n \choose j} \check{P} (\underbrace{x,x,\dots ,x}_{j} & \underbrace{y,y,\dots ,y}_{n-j})

However, in order to apply (and understand) this property you must know how to calculate \begin{pmatrix} n \\ j \end{pmatrix} , called binomial coefficient. So if you don’t understand the previous property, we recommend that you see what the formula for the binomial coefficient is .

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