mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a diagonal form is an algebraic form ( homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is :

\sum_^n a_i ^m\

for some given degree ''m''. Such forms ''F'', and the hypersurfaces ''F'' = 0 they define in

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

, are very special in geometric terms, with many symmetries. They also include famous cases like the Fermat curves, and other examples well known in the theory of

Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...

s. A great deal has been worked out about their theory:

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, local zeta-functions via

Jacobi sum In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums ''J''(''χ'', ''ψ'') for Dirichlet characters ''χ'', ''ψ'' modulo a prime number ''p'', defined by : J(\chi,\psi) = ...

s, Hardy-Littlewood circle method.

Examples

X^2+Y^2-Z^2 = 0

is the unit circle in ''P''² :

X^2-Y^2-Z^2 = 0

is the unit hyperbola in ''P''². :

x_0^3+x_1^3+x_2^3+x_3^3=0

gives the Fermat cubic surface in ''P''³ with 27 lines. The 27 lines in this example are easy to describe explicitly: they are the 9 lines of the form (''x'' : ''ax'' : ''y'' : ''by'') where ''a'' and ''b'' are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates. :

x_0^4+x_1^4+x_2^4+x_3^4=0

gives a K3 surface in ''P''³. {{DEFAULTSORT:Diagonal Form Homogeneous polynomials Algebraic varieties