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Polynomials in Many Variables

Let be an arbitrary field. A monomial in variables is a product of the form , where all the exponents are non-negative integers. The total degree of this monomial is the sum .

Often the following simplified notation is used: if be an -tuple of non-negative integers, then we write . For instance, if , then . We also write for the total degree of the monomial .

A polynomial in variables with coefficients in is a finite linear combination with coefficients in of monomials in variables . A polynomial can be written in the form

(1)

where the sum is over a finite number of -tuples .

The set of all polynomials in variables with coefficients in is denoted by .

Let in equation (1) be a polynomial in . We call the coefficient of the monomial . If the coefficient is non-vanishing, then we call a term of . The total degree of , denoted by , is the maximum such that the coefficient is non-zero.

The polynomial given in equation (1) is called the zero polynomial provided all of its coefficients vanish.

A polynomial in equation (1) defines a function in a natural way as follows: given , replace every by in the expression of . The result is an element of . This gives the equation two potential meanings:

• is the zero polynomial, that is all of its coefficients are vanishing, or
• if the zero function, that is for all ?

This two statements are not equivalent in general. If , the field with two elements , and then defines the zero function but it is not the zero polynomial. More generally, if is a prime number then for all , and consequently for all .

However, when is infinite, there is no problem with this duality:

Theorem: Let be an infinite field, and let . Then is the zero polynomial in if and only if is the zero function.

Corollary: Let be an infinite field, and let . Then in if and only if and are identical as the functions defined on .

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