Editing Quadric (section)

==Rational points==
A quadric is ''defined over'' a [[field (mathematics)|field]] <math>F</math> if the coefficients of its equation belong to <math>F.</math> When <math>F</math> is the field <math>\Q</math> of the [[rational number]]s, one can suppose that the coefficients are [[integer]]s by [[clearing denominators]].

A point of a quadric defined over a field <math>F</math> is said [[rational point|rational]] over <math>F</math> if its coordinates belong to <math>F.</math> A rational point over the field <math>\R</math> of the real numbers, is called a real point.

A rational point over <math>\Q</math> is called simply a ''rational point''. By clearing denominators, one can suppose and one supposes generally that the [[projective coordinates]] of a rational point (in a quadric defined over <math>\Q</math>) are integers. Also, by clearing denominators of the coefficients, one supposes generally that all the coefficients of the equation of the quadric and the polynomials occurring in the parametrization are integers.

Finding the rational points of a projective quadric amounts thus to solving a [[Diophantine equation]].

Given a rational point {{mvar|A}} over a quadric over a field {{mvar|F}}, the parametrization described in the preceding section provides rational points when the parameters are in {{mvar|F}}, and, conversely, every rational point of the quadric can be obtained from parameters in {{mvar|F}}, if the point is not in the tangent hyperplane at {{mvar|A}}.

It follows that, if a quadric has a rational point, it has many other rational points (infinitely many if {{mvar|F}} is infinite), and these points can be algorithmically generated as soon one knows one of them.

As said above, in the case of projective quadrics defined over <math>\Q,</math> the parametrization takes the form
:<math>X_i=F_i(T_1, \ldots, T_n)\quad \text{for } i=0,\ldots,n,</math>
where the <math>F_i</math> are homogeneous polynomials of degree two with integer coefficients. Because of the homogeneity, one can consider only parameters that are [[setwise coprime]] integers. If <math>Q(X_0,\ldots, X_n)=0</math> is the equation of the quadric, a solution of this equation is said ''primitive'' if its components are setwise coprime integers. The primitive solutions are in one to one correspondence with the rational points of the quadric ([[up to]] a change of sign of all components of the solution). The non-primitive integer solutions are obtained by multiplying primitive solutions by arbitrary integers; so they do not deserve a specific study. However, setwise coprime parameters can produce non-primitive solutions, and one may have to divide by a [[greatest common divisor]] to arrive at the associated primitive solution.

===Pythagorean triples===
This is well illustrated by [[Pythagorean triple]]s. A Pythagorean triple is a [[triple (mathematics)|triple]] <math>(a,b,c)</math> of positive integers such that <math>a^2+b^2=c^2.</math> A Pythagorean triple is ''primitive'' if <math>a, b, c</math> are setwise coprime, or, equivalently, if any of the three pairs <math>(a,b),</math> <math>(b,c)</math> and <math>(a,c)</math> is coprime.

By choosing <math>A=(-1, 0, 1),</math> the above method provides the parametrization
:<math>\begin{cases}
a=m^2-n^2\\b=2mn\\c=m^2+n^2 
\end{cases}</math> 
for the quadric of equation <math>a^2+b^2-c^2=0.</math> (The names of variables and parameters are being changed from the above ones to those that are common when considering Pythagorean triples).

If {{mvar|m}} and {{mvar|n}} are coprime integers such that <math>m>n>0,</math>  the resulting triple is a Pythagorean triple. If one of {{mvar|m}} and {{mvar|n}} is even and the other is odd, this resulting triple is primitive; otherwise, {{mvar|m}} and {{mvar|n}} are both odd, and one obtains a primitive triple by dividing by 2.

In summary, the primitive Pythagorean triples with <math>b</math> even are obtained as
:<math>a=m^2-n^2,\quad b=2mn,\quad c= m^2+n^2,</math> 
with {{mvar|m}} and {{mvar|n}} coprime integers such that one is even and <math>m>n>0</math> (this is [[Euclid's formula]]). The primitive Pythagorean triples with <math>b</math> odd are obtained as
:<math>a=\frac{m^2-n^2}{2},\quad b=mn, \quad c= \frac{m^2+n^2}2,</math>
with {{mvar|m}} and {{mvar|n}} coprime odd integers such that <math>m>n>0.</math>

As the exchange of {{mvar|a}} and {{mvar|b}} transforms a Pythagorean triple into another Pythagorean triple, only one of the two cases is sufficient for producing all primitive Pythagorean triples [[up to]] the order of {{mvar|a}} and {{mvar|b}}.