A polynomial in an indeterminate is an expression of the form , where the are called the coefficients of the polynomial. Two such polynomials are equal only if the corresponding coefficients are equal.[4] In contrast, two polynomial functions in a variable may be equal or not at a particular value of .
For example, the functions
are equal when and not equal otherwise. But the two polynomials
are unequal, since 2 does not equal 5, and 3 does not equal 2. In fact,
does not hold unless and . This is because is not, and does not designate, a number.
The distinction is subtle, since a polynomial in can be changed to a function in by substitution. But the distinction is important because information may be lost when this substitution is made. For example, when working in modulo 2, we have that:
so the polynomial function is identically equal to 0 for having any value in the modulo-2 system. However, the polynomial is not the zero polynomial, since the coefficients, 0, 1 and −1, respectively, are not all zero.
Indeterminates are useful in abstract algebra for generating mathematical structures. For example, given a field , the set of polynomials with coefficients in is the polynomial ring with polynomial addition and multiplication as operations. In particular, if two indeterminates and are used, then the polynomial ring also uses these operations, and convention holds that .
Indeterminates may also be used to generate a free algebra over a commutative ring . For instance, with two indeterminates and , the free algebra includes sums of strings in and , with coefficients in , and with the understanding that and are not necessarily identical (since free algebra is by definition non-commutative).
Weisstein, Eric W. "Indeterminate". mathworld.wolfram.com. Retrieved 2019-12-02.