Editing Equation (section)

==Number theory==

===Diophantine equations===
{{main|Diophantine equation}}
A Diophantine equation is a [[polynomial equation]] in two or more unknowns for which only the [[integer]] [[Zero of a function#Polynomial roots|solutions]] are sought (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of [[monomials]] of [[Degree of a polynomial|degree]] zero or one. An example of linear Diophantine equation is {{math|''ax'' + ''by'' {{=}} ''c''}} where ''a'', ''b'', and ''c'' are constants. An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns.

Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an [[algebraic curve]], [[algebraic surface]], or more general object, and ask about the [[lattice point]]s on it.

The word ''Diophantine'' refers to the [[Greek mathematics#Hellenistic|Hellenistic mathematician]] of the 3rd century, [[Diophantus]] of [[Alexandria]], who made a study of such equations and was one of the first mathematicians to introduce [[Mathematical symbol|symbolism]] into [[algebra]]. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.

===Algebraic and transcendental numbers===
{{main|Algebraic number|Transcendental number}}
An [[algebraic number]] is a number that is a solution of a non-zero [[polynomial equation]] in one variable with [[rational number|rational]] coefficients (or equivalently — by [[clearing denominators]] — with [[integer]] coefficients). Numbers such as [[Pi|{{pi}}]] that are not algebraic are said to be [[transcendental number|transcendental]]. [[Almost all]] [[real number|real]] and [[complex number|complex]] numbers are transcendental.

===Algebraic geometry===
{{main|Algebraic geometry}}
[[Algebraic geometry]] is a branch of [[mathematics]], classically studying solutions of [[polynomial equations]]. Modern algebraic geometry is based on more abstract techniques of [[abstract algebra]], especially [[commutative algebra]], with the language and the problems of [[geometry]].

The fundamental objects of study in algebraic geometry are [[algebraic variety|algebraic varieties]], which are geometric manifestations of [[solution set|solutions]] of [[systems of polynomial equations]]. Examples of the most studied classes of algebraic varieties are: [[plane algebraic curve]]s, which include [[line (geometry)|lines]], [[circle]]s, [[parabola]]s, [[ellipse]]s, [[hyperbola]]s, [[cubic curve]]s like [[elliptic curve]]s and quartic curves like [[lemniscate of Bernoulli|lemniscates]], and [[Cassini oval]]s. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the [[singular point of a curve|singular points]], the [[inflection point]]s and the [[point at infinity|points at infinity]]. More advanced questions involve the [[topology]] of the curve and relations between the curves given by different equations.