Indeterminate (variable)

In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else except itself. It may be 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. It is not a variable designating a function argument, or a variable being summed or integrated over. It is not any type of bound variable. It is just a symbol used in an entirely formal way. When used as placeholders, a common operation is to substitute mathematical expressions (of an appropriate type) for the indeterminates. By a common abuse of language, mathematical texts may not clearly distinguish indeterminates from ordinary variables. Polynomial 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. 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. Formal power series A formal power series in an indeterminate is an expression of the form , where no value is assigned to the symbol . This is similar to the definition of a polynomial, except that an infinite number of the coefficients may be nonzero.