numbers curve algebra surface curves
The rephrasing of questions on equations in terms of points on curves turns out to be felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve—that is, rational or integer solutions to an equation 



f
(
x
,
y
)
=
0


{\displaystyle f(x,y)=0}

, where 



f


{\displaystyle f}

 is a polynomial in two variables—turns out to depend crucially on the genus of the curve. The genus can be defined as follows: allow the variables in 



f
(
x
,
y
)
=
0


{\displaystyle f(x,y)=0}

 to be complex numbers; then 



f
(
x
,
y
)
=
0


{\displaystyle f(x,y)=0}

 defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables, that is, four dimensions). If we count the number of (doughnut) holes in the surface; we call this number the genus of 



f
(
x
,
y
)
=
0


{\displaystyle f(x,y)=0}

. Other geometrical notions turn out to be just as crucial.
