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The word "pole" is used prominently in a number of very different branches of mathematics. Perhaps the most important and widespread usage is to denote a singularity of a ...

An equation of the form P(x)=0, where P(x) is a polynomial.

A homogeneous polynomial in two or more variables.

If a polynomial P(x) is divided by (x-r), then the remainder is a constant given by P(r).

The ring R[x] of polynomials in a variable x.

A sequence of polynomials p_i(x), for i=0, 1, 2, ..., where p_i(x) is exactly of degree i for all i.

The difference between the highest and lowest degrees of a polynomial.

If an n-sided Poncelet transverse constructed for two given conic sections is closed for one point of origin, it is closed for any position of the point of origin. ...

The relationship x precedes y is written x≺y. The relation x precedes or is equal to y is written x<=y.

alpha is called a predecessor if there is no ordinal number beta such that beta+1=alpha.