mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a diagonal form is an algebraic form (

homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables ...

) without cross-terms involving different indeterminates. That is, it is of the form :

\sum_^n a_i ^m\

for some degree ''m''. Such forms ''F'', and the

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...

s ''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 curve In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (''X'':''Y'':''Z'') by the Fermat equation: :X^n + Y^n = Z^n.\ Therefore, in terms of the affine plane its equation is: ...

s, and other examples well known in the theory of

Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...

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

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

local zeta-function In mathematics, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as :Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^k\right) where is a non-singular -dimensional projective algeb ...

s via Jacobi sums, Hardy-Littlewood circle method.

Diagonalization

Any degree-2 homogeneous polynomial can be transformed to a diagonal form by variable substitution. Higher-degree homogeneous polynomials can be diagonalized if and only if their catalecticant is non-zero. The process is particularly simple for degree-2 forms (

quadratic forms In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...

), based on the

eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

of the symmetric matrix representing the quadratic form.

Examples

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

is the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

in ''P''² :

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

is the

unit hyperbola In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative rad ...

in ''P''². :

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

gives the Fermat

cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than ...

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 mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity of a surface, irregularity zero. An (algebraic) K3 surface over any field (mathematics), field ...

in ''P''³.

References

{{DEFAULTSORT:Diagonal Form Homogeneous polynomials Algebraic varieties