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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:
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
.
Cite this web-page as:
Štefan Porubský: Polynomials in Many Variables.