Discriminant

The discriminant ( Latin discriminare = differentiate) is a mathematical expression that allows statements about the number and type of solutions to an algebraic equation . The best known is the discriminant of a quadratic equation .

Discriminant of a quadratic equation

The solutions (roots) of a quadratic equation with real coefficients , and can be expressed using the midnight formula ${\ displaystyle ax ^ {2} + bx + c = 0}$ ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle c}$

{\ displaystyle x_ {1,2} = {\ frac {-b \ pm {\ sqrt {b ^ {2} -4ac}}} {2a}}}

to calculate. The number of real solutions depends on the term under the root, the so-called radicand.

This expression

{\ displaystyle b ^ {2} -4ac}

is called the discriminant of the quadratic equation and is referred to below with . ${\ displaystyle ax ^ {2} + bx + c = 0}$ ${\ displaystyle D}$

For the square root in the solution formula has a positive value, so there are two different real solutions and . ${\ displaystyle D> 0}$ ${\ displaystyle x_ {1}}$ ${\ displaystyle x_ {2}}$
For , the square root has the value 0. Since it makes no difference whether you add or subtract 0, there is exactly one real solution (the multiplicity 2) despite the plus-minus sign. ${\ displaystyle D = 0}$
For , the square root of the solution formula in the field of real numbers ( ) is not defined. So there is no real solution. The situation is different if one takes the body of the complex numbers as a basis. In this case there are two (non-real) solutions that are complex conjugate to each other . ${\ displaystyle D <0}$ ${\ displaystyle \ mathbb {R}}$

Motivation of the general term discriminant

Let it be a polynomial with zeros , some of which may be complex. The expression ${\ displaystyle p_ {n} = a_ {n} x ^ {n} + a_ {n-1} x ^ {n-1} + \ dotsb + a_ {1} x + a_ {0} \ in \ mathbb { R} [x]}$ ${\ displaystyle x_ {1}, x_ {2}, \ dotsc, x_ {n}}$

{\ displaystyle (x_ {1} -x_ {2}) (x_ {1} -x_ {3}) \ dotsm (x_ {2} -x_ {3}) (x_ {2} -x_ {4}) \ dotsm (x_ {3} -x_ {4}) \ dotsm (x_ {n-1} -x_ {n}) = \ prod _ {i <j} (x_ {i} -x_ {j})}

,

which consists of factors (one factor for each pair of zeros) disappears if (at least) one zero occurs more than once. The expression is not symmetric about the zeros, i. This means that its value may change if the zeros are renumbered. The symmetry can be enforced by squaring all factors: ${\ displaystyle {\ tbinom {n} {2}}}$

{\ displaystyle D_ {n} = a_ {n} ^ {2n-2} (x_ {1} -x_ {2}) ^ {2} (x_ {1} -x_ {3}) ^ {2} \ dotsm (x_ {2} -x_ {3}) ^ {2} (x_ {2} -x_ {4}) ^ {2} \ dotsm (x_ {3} -x_ {4}) ^ {2} \ dotsm ( x_ {n-1} -x_ {n}) ^ {2}}

.

This expression is a homogeneous symmetric polynomial of degree . It is called the discriminant of the polynomial . (The meaning of the normalization term is explained below.) ${\ displaystyle D_ {n}}$ ${\ displaystyle n (n-1)}$ ${\ displaystyle p_ {n}}$ ${\ displaystyle a_ {n} ^ {2n-2}}$

Examples

Quadratic polynomial

A general polynomial of degree 2 has the form with . Its discriminant is . ${\ displaystyle p_ {2} = ax ^ {2} + bx + c}$ ${\ displaystyle a \ neq 0}$ ${\ displaystyle D_ {2} = a ^ {2} (x_ {1} -x_ {2}) ^ {2} = a ^ {2} (x_ {1} ^ {2} + x_ {2} ^ { 2} -2x_ {1} x_ {2})}$

With the set of Vieta and completing the square , they can be transformed into: . ${\ displaystyle D_ {2} = a ^ {2} \ left ((x_ {1} + x_ {2}) ^ {2} -4x_ {1} x_ {2} \ right) = a ^ {2} \ left (\ left ({\ frac {-b} {a}} \ right) ^ {2} -4 {\ frac {c} {a}} \ right) = b ^ {2} -4ac}$

The quadratic polynomial has a double zero if and only if applies. ${\ displaystyle p_ {2}}$ ${\ displaystyle b ^ {2} -4ac = 0}$

Cubic polynomial

A general polynomial of degree 3 has the form with . Its discriminant is . ${\ displaystyle p_ {3} = ax ^ {3} + bx ^ {2} + cx + d}$ ${\ displaystyle a \ neq 0}$ ${\ displaystyle D_ {3} = a ^ {4} (x_ {1} -x_ {2}) ^ {2} (x_ {1} -x_ {3}) ^ {2} (x_ {2} -x_ {3}) ^ {2}}$

With Vieta's theorem it can (with a complex calculation) be transformed into

{\ displaystyle D_ {3} = b ^ {2} c ^ {2} -4ac ^ {3} -4b ^ {3} d + 18abcd-27a ^ {2} d ^ {2}}

.

This expression is unwieldy and difficult to remember. If you bring the polynomial to the form with a similar addition as with quadratic addition, the formula is easier to remember: ${\ displaystyle x ^ {3} + ax + b}$ ${\ displaystyle D_ {3} = - 4a ^ {3} -27b ^ {2}}$

Taking into consideration, that each cubic equation after division by and subsequent substitution on an equation of the form can be brought, one obtains a more noticeable for the discriminant formula: . ${\ displaystyle ax ^ {3} + bx ^ {2} + cx + d = 0}$ ${\ displaystyle a}$ ${\ displaystyle y = x + {\ tfrac {b} {3a}}}$ ${\ displaystyle y ^ {3} + 3py + 2q = 0}$ ${\ displaystyle D_ {3} = - 108a ^ {4} (p ^ {3} + q ^ {2})}$

A reduced cubic polynomial has a multiple zero if and only if applies. In school books, this term is often referred to as discriminant, so the factor is ignored. ${\ displaystyle p_ {3} = y ^ {3} + 3py + 2q}$ ${\ displaystyle p ^ {3} + q ^ {2} = 0}$ ${\ displaystyle -108a ^ {4}}$

Higher Degree Polynomials

The procedure described above works for polynomials of any degree. From the theory of symmetric polynomials and the Vieta theorem it follows that the expression

{\ displaystyle D_ {n} = a_ {n} ^ {2n-2} (x_ {1} -x_ {2}) ^ {2} (x_ {1} -x_ {3}) ^ {2} \ dotsm (x_ {2} -x_ {3}) ^ {2} (x_ {2} -x_ {4}) ^ {2} \ dotsm (x_ {3} -x_ {4}) ^ {2} \ dotsm ( x_ {n-1} -x_ {n}) ^ {2}}

can always be represented in a unique way as a (polynomial) function of the coefficients of the polynomial . ${\ displaystyle p_ {n}}$

Comments on the sign of the discriminant

If all zeros of a polynomial are real, then the discriminant is Das follows immediately from the definition. ${\ displaystyle D \ geq 0.}$
The converse also applies to quadratic and cubic polynomials: If all zeros are real. ${\ displaystyle D \ geq 0,}$
The polynomial has four zeros , , and . The discriminant has the value 16384, so it is positive. However, the zeros are not real. ${\ displaystyle p_ {4} = x ^ {4} +4}$ ${\ displaystyle 1 + i}$ ${\ displaystyle 1-i}$ ${\ displaystyle -1 + i}$ ${\ displaystyle -1-i}$

Normalization factor

In the definition used above, the factor occurs. It has the effect that when using Vieta's theorem, the denominators disappear, i.e. the discriminant appears as a polynomial in the coefficients . Depending on the context and purpose of the discriminant, the definition is slightly modified: ${\ displaystyle a_ {n} ^ {2n-2}}$ ${\ displaystyle a_ {0}, a_ {1}, \ dotsc, a_ {n}}$

The factor is set instead of . ${\ displaystyle a_ {n} ^ {2n-2}}$ ${\ displaystyle (-1) ^ {n (n-1) / 2} a_ {n} ^ {2n-2}}$
The factor is set instead of . ${\ displaystyle a_ {n} ^ {2n-2}}$ ${\ displaystyle (-1) ^ {n (n-1) / 2} a_ {n} ^ {2n-1}}$
The factor is set instead of . ${\ displaystyle a_ {n} ^ {2n-2}}$ ${\ displaystyle a_ {n} ^ {2n-1}}$
The factor is omitted. ${\ displaystyle a_ {n} ^ {2n-2}}$

With the first three variants, caution is advised with statements like those made in the section “Comments on the sign of the discriminant”.

general definition

Let be a univariate polynomial (i.e. a polynomial in an unknown) over a commutative unitary ring . The discriminant of is defined as the resultant of with its derivative reduced by : ${\ displaystyle f = f_ {0} + f_ {1} X + \ dotsb + f_ {n} X ^ {n} \ in R [X]}$ ${\ displaystyle f}$ ${\ displaystyle f_ {n}}$ ${\ displaystyle f}$ ${\ displaystyle f '}$

{\ displaystyle f_ {n} \ operatorname {Disk} (f) = (- 1) ^ {n (n-1) / 2} \ operatorname {Res} (f, f ')}

.

The discriminant is also indicated by the symbol . ${\ displaystyle \ Delta (f)}$

If a body is and , then applies as above ${\ displaystyle R = K}$ ${\ displaystyle f_ {n} = 1}$

{\ displaystyle \ operatorname {Disk} (f) = \ prod _ {i <j} (x_ {i} -x_ {j}) ^ {2}}

;

where the zeros of are in an algebraic closure of . ${\ displaystyle x_ {1}, \ dotsc, x_ {n}}$ ${\ displaystyle f}$ ${\ displaystyle K}$

Note: Often the discriminant is defined without the additional factor ; the corresponding prefactor must then be added to the formula given above to calculate the discriminant from the zeros. ${\ displaystyle (-1) ^ {n (n-1) / 2}}$

comment

Written out the resultant of a polynomial with its derivative is equal to the determinant of the - matrix . ${\ displaystyle f (x) = f_ {0} + f_ {1} x + \ dotsb + f_ {n} x ^ {n}}$ ${\ displaystyle f '(x) = f_ {1} + \ dotsb + nf_ {n} x ^ {n-1}}$ ${\ displaystyle (2n-1) \ times (2n-1)}$