Editing Discriminant (section)

==Properties==
===Zero discriminant===
The discriminant of a polynomial over a [[field (mathematics)|field]] is zero if and only if the polynomial has a multiple root in some [[field extension]].

The discriminant of a polynomial over an [[integral domain]] is zero if and only if the polynomial and its [[formal derivative|derivative]] have a non-constant common divisor.

In [[characteristic (algebra)|characteristic]] 0, this is equivalent to saying that the polynomial is not [[square-free polynomial|square-free]] (i.e., it is divisible by the square of a non-constant polynomial).

In nonzero characteristic {{math|''p''}}, the discriminant is zero if and only if the polynomial is not square-free or it has an [[irreducible polynomial|irreducible factor]] which is not separable (i.e., the irreducible factor is a polynomial in <math>x^p</math>).

===Invariance under change of the variable===
The discriminant of a polynomial is, [[up to]] a scaling, invariant under any [[projective transformation]] of the variable. As a projective transformation may be decomposed into a product of translations, homotheties and inversions, this results in the following formulas for simpler transformations, where {{math|''P''(''x'')}} denotes a polynomial of degree {{math|''n''}}, with <math>a_n</math> as leading coefficient.

* ''Invariance by translation'':
::<math>\operatorname{Disc}_x(P(x+\alpha)) = \operatorname{Disc}_x(P(x))</math>
:This results from the expression of the discriminant in terms of the roots
* ''Invariance by homothety'':
::<math>\operatorname{Disc}_x(P(\alpha x)) = \alpha^{n(n-1)}\operatorname{Disc}_x(P(x))</math>
:This results from the expression in terms of the roots, or of the quasi-homogeneity of the discriminant.
* ''Invariance by inversion'':
::<math>\operatorname{Disc}_x(P^{\mathrm{r}}\!\!\;(x)) = \operatorname{Disc}_x(P(x))</math>
:when <math>P(0)\ne 0.</math> Here, <math>P^{\mathrm{r}}\!\!\;</math> denotes the [[reciprocal polynomial]] of {{math|''P''}}; that is, if <math>P(x) = a_nx^n + \cdots + a_0,</math> and  <math>a_0 \neq 0,</math>  then
::<math>P^{\mathrm{r}}\!\!\;(x) = x^nP(1/x) = a_0x^n +\cdots +a_n.</math>

===Invariance under ring homomorphisms===
Let <math>\varphi\colon R \to S</math> be a [[ring homomorphism|homomorphism]] of [[commutative ring]]s. Given a polynomial 
:<math>A = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0</math>
in {{math|''R''[''x'']}}, the homomorphism <math>\varphi</math> acts on {{math|''A''}} for producing the polynomial
:<math>A^\varphi = \varphi(a_n)x^n+\varphi(a_{n-1})x^{n-1}+ \cdots+\varphi(a_0)</math>
in {{math|''S''[''x'']}}.

The discriminant is invariant under <math>\varphi</math> in the following sense. If <math>\varphi(a_n)\ne 0,</math> then 
:<math>\operatorname{Disc}_x(A^\varphi) = \varphi(\operatorname{Disc}_x(A)).</math>
As the discriminant is defined in terms of a determinant, this property results immediately from the similar property of determinants.

If <math>\varphi(a_n)= 0,</math> then <math>\varphi(\operatorname{Disc}_x(A))</math> may be zero or not. One has, when <math>\varphi(a_n)= 0,</math>
:<math>\varphi(\operatorname{Disc}_x(A)) = \varphi(a_{n-1})^2\operatorname{Disc}_x(A^\varphi).</math>

When one is only interested in knowing whether a discriminant is zero (as is generally the case in [[algebraic geometry]]), these properties may be summarised as:
:<math>\varphi(\operatorname{Disc}_x(A)) = 0</math> if and only if either <math>\operatorname{Disc}_x(A^\varphi)=0</math> or <math>\deg(A)-\deg(A^\varphi)\ge 2.</math>

This is often interpreted as saying that <math>\varphi(\operatorname{Disc}_x(A)) = 0</math> if and only if <math>A^\varphi</math> has a [[multiple root]] (possibly [[point at infinity|at infinity]]).

===Product of polynomials===
If {{math|1=''R'' = ''PQ''}} is a product of polynomials in {{math|''x''}}, then
:<math>\begin{align}
\operatorname{disc}_x(R) &= \operatorname{disc}_x(P)\operatorname{Res}_x(P,Q)^2\operatorname{disc}_x(Q)
\\[5pt]
{}&=(-1)^{pq}\operatorname{disc}_x(P)\operatorname{Res}_x(P,Q)\operatorname{Res}_x(Q,P)\operatorname{disc}_x(Q),
\end{align}</math>
where <math>\operatorname{Res}_x</math> denotes the [[resultant]] with respect to the variable {{math|''x''}}, and {{math|''p''}} and {{math|''q''}} are the respective degrees of {{math|''P''}} and {{math|''Q''}}.

This property follows immediately by substituting the expression for the resultant, and the discriminant, in terms of the roots of the respective polynomials.

===Homogeneity===
The discriminant is a [[homogeneous polynomial]] in the coefficients; it is also a homogeneous polynomial in the roots and thus [[quasi-homogeneous polynomial|quasi-homogeneous]] in the coefficients.

The discriminant of a polynomial of degree {{math|''n''}} is homogeneous of degree {{math|2''n'' − 2}} in the coefficients. This can be seen in two ways. In terms of the roots-and-leading-term formula, multiplying all the coefficients by {{mvar|λ}} does not change the roots, but multiplies the leading term by {{mvar|λ}}. In terms of its expression as a determinant of a {{math|(2''n'' − 1)&thinsp;×&thinsp;(2''n'' − 1)}} [[matrix (mathematics)|matrix]] (the [[Sylvester matrix]]) divided by {{mvar|a<sub>n</sub>}}, the determinant is homogeneous of degree {{math|2''n'' − 1}} in the entries, and dividing by {{mvar|a<sub>n</sub>}} makes the degree {{math|2''n'' − 2}}.

The discriminant of a polynomial of degree {{math|''n''}} is homogeneous of degree {{math|''n''(''n'' − 1)}} in the roots. This follows from the expression of the discriminant in terms of the roots, which is the product of a constant and <math>\binom{n}{2} = \frac{n(n-1)}{2}</math> squared differences of roots.

The discriminant of a polynomial of degree {{math|''n''}} is quasi-homogeneous of degree {{math|''n''(''n'' − 1)}} in the coefficients, if, for every {{math|''i''}}, the coefficient of <math>x^i</math> is given the weight {{math|''n'' − ''i''}}. It is also quasi-homogeneous of the same degree, if, for every {{math|''i''}}, the coefficient of <math>x^i</math> is given the weight {{math|''i''}}. This is a consequence of the general fact that every polynomial which is homogeneous and [[symmetric polynomial|symmetric]] in the roots may be expressed as a quasi-homogeneous polynomial in the [[elementary symmetric function]]s of the roots.

Consider the polynomial
:<math> P=a_nx^n+a_{n-1}x^{n-1}+ \cdots +a_0.</math>
It follows from what precedes that the exponents in every [[monomial]] <math>a_0^{i_0}, \dots , a_n^{i_n}</math> appearing in the discriminant satisfy the two equations
:<math>i_0+i_1+\cdots+i_n=2n-2</math>
and
:<math>i_1+2i_2 + \cdots+n i_n=n(n-1),</math>
and also the equation
:<math>ni_0 +(n-1)i_1+ \cdots+ i_{n-1}=n(n-1),</math>
which is obtained by subtracting the second equation from the first one multiplied by {{math|''n''}}.

This restricts the possible terms in the discriminant. For the general quadratic polynomial, the discriminant <math>b^2-4ac</math> is a homogeneous polynomial of degree 2 which has only two there are only two terms, while the general homogeneous polynomial of degree two in three variables has 6 terms. The discriminant of the  general cubic polynomial is a homogeneous polynomial of degree 4 in four variables; it has five terms, which is the maximum allowed by the above rules, while the general homogeneous polynomial of degree 4 in 4 variables has 35 terms.

For higher degrees, there may be monomials which satisfy above rules and do not appear in the discriminant. The first example is for the quartic polynomial <math>ax^4 + bx^3 + cx^2 + dx + e</math>, in which case the monomial <math>bc^4d</math> satisfies the rules without appearing in the discriminant.