Indeterminate (variable)

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In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, does not stand for anything else except itself, and is often used as a placeholder in objects such as polynomials and formal power series.^[1][2] 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.^[3]

Polynomials[]

A polynomial in an indeterminate ${\displaystyle X}$ is an expression of the form ${\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\ldots +a_{n}X^{n}}$, where the ${\displaystyle a_{i}}$ 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 ${\displaystyle x}$ may be equal or not at a particular value of ${\displaystyle x}$.

For example, the functions

${\displaystyle f(x)=2+3x,\quad g(x)=5+2x}$

are equal when ${\displaystyle x=3}$ and not equal otherwise. But the two polynomials

${\displaystyle 2+3X,\quad 5+2X}$

are unequal, since 2 does not equal 5, and 3 does not equal 2. In fact,

${\displaystyle 2+3X=a+bX}$

does not hold unless ${\displaystyle a=2}$ and ${\displaystyle b=3}$. This is because ${\displaystyle X}$ is not, and does not designate, a number.

The distinction is subtle, since a polynomial in ${\displaystyle X}$ can be changed to a function in ${\displaystyle x}$ 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:

${\displaystyle 0-0^{2}=0,\quad 1-1^{2}=0,}$

so the polynomial function ${\displaystyle x-x^{2}}$ is identically equal to 0 for ${\displaystyle x}$ having any value in the modulo-2 system. However, the polynomial ${\displaystyle X-X^{2}}$ 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 ${\displaystyle X}$ is an expression of the form ${\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\ldots }$, where no value is assigned to the symbol ${\displaystyle X}$.^[5] 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 (since there is no function at play). So power series that would diverge for values of ${\displaystyle x}$, such as ${\displaystyle 1+x+2x^{2}+6x^{3}+\ldots +n!x^{n}+\ldots \,}$, are allowed.

As generators[]

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

Indeterminates may also be used to generate a free algebra over a commutative ring ${\displaystyle A}$. For instance, with two indeterminates ${\displaystyle X}$ and ${\displaystyle Y}$, the free algebra ${\displaystyle A\langle X,Y\rangle }$ includes sums of strings in ${\displaystyle X}$ and ${\displaystyle Y}$, with coefficients in ${\displaystyle A}$, and with the understanding that ${\displaystyle XY}$ and ${\displaystyle YX}$ are not necessarily identical (since free algebra is by definition non-commutative).

See also[]

• Indeterminate equation
• Indeterminate form
• Indeterminate system
• Polynomial
• Formal power series

Notes[]

References[]

• Herstein, I. N. (1975). Topics in Algebra. Wiley.
• McCoy, Neal H. (1973), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225

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