Editing Discriminant (section)

===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.