Inmathematics, adiagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving differentindeterminates. That is, it is of the form

${\displaystyle \sum _{i=1}^n{a}_i{x_i}^m}$

for some degreem.

Such formsF, and thehypersurfacesF = 0 they define inprojective space, are very special in geometric terms, with many symmetries. They also include famous cases like theFermat curves, and other examples well known in the theory ofDiophantine equations.

A great deal has been worked out about their theory:algebraic geometry,local zeta-functions viaJacobi sums,Hardy-Littlewood circle method.

Any degree-2 homogeneous polynomial can be transformed to a diagonal form by variable substitution.^[1] Higher-degree homogeneous polynomials can be diagonalized if and only if theircatalecticant is non-zero.

The process is particularly simple for degree-2 forms (quadratic forms), based on theeigenvalues of the symmetric matrix representing the quadratic form.

${\displaystyle X^2+Y^2-Z^2=0}$  is theunit circle inP²

${\displaystyle X^2-Y^2-Z^2=0}$  is theunit hyperbola inP².

${\displaystyle x_0^3+x_1^3+x_2^3+x_3^3=0}$  gives the Fermatcubic surface inP³ 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) wherea andb are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates.

${\displaystyle x_0^4+x_1^4+x_2^4+x_3^4=0}$  gives aK3 surface inP³.