Indeterminate (variable)

In mathematics, and particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else but itself and is used as a placeholder in objects such as polynomials and formal power series. In particular, it does not designate a constant or a parameter of the problem, it is not an unknown that could be solved for, and it is not a variable designating a function argument or being summed or integrated over; it is not any type of bound variable.

Polynomials

A polynomial in an indeterminate X is an expression of the form $a_+a_X+a_X^+\ldots +a_X^$ , where the ai are called the coefficients of the polynomial. Two such polynomials are equal only if the corresponding coefficients are equal. In contrast, two polynomial functions in a variable x may be equal or not depending on the value of x.

For example, the functions

$f(x)=2+3x,\quad g(x)=5+2x$ are equal when x = 3 and not equal otherwise. But the two polynomials

$2+3X,\quad 5+2X$ are unequal, since 2 does not equal 5, and 3 does not equal 2. In fact,

$2+3X=a+bX$ does not hold unless a = 2 and b = 3. This is because X is not, and does not designate, a number.

The distinction is subtle, since a polynomial in X can be changed to a function in x by substitution. But the distinction is important because information may be lost when this substitution is made. Working in modulo 2:

$0-0^=0,\quad 1-1^=0,$ so the polynomial function xx2 is identically equal to 0 for x having any value in the modulo-2 system. But the polynomial XX2 is not the zero polynomial, since the coefficients, 0, 1 and −1, are not all zero.

Formal power series

A formal power series in an indeterminate X is an expression of the form a0 + a1X + a2X2 + …. This is similar to the definition of a polynomial, except that an infinite number of the coefficients may be nonzero. Unlike the power series encountered in calculus, questions of convergence are irrelevant. So power series that would diverge for values of x, such as 1 + x + 2x2 + 6x3 + … + n!xn + …, are allowed.

As generators

Indeterminates are useful in abstract algebra for generating mathematical structures. For example, given a field K, the set of polynomials with coefficients in K is the polynomial ring with polynomial addition and multiplication as operations. If two indeterminates X and Y are used, the polynomial ring K[X,Y] also uses these operations, and convention holds that XY = YX.

Indeterminates may also be used to generate a free algebra over a commutative ring A. For instance, with two indeterminates X and Y, the free algebra AX,Y⟩ includes sums of strings in X and Y, with coefficients in A, and with the understanding that XY and YX are not necessarily identical.