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II. The Algebra Related to Cubic Equation

The central problem related to equation (1) is, of course, to find its roots. There are many methods for the general solution of this problem, conveniently coping with the purpose that solution is serving (Cocolicchio, Viggiano, 2000; Nickalls, 1993). All of the methods of solution are however centered on reducing it to that of a quadratic equation and, for Physics and Engineering purposes, we think it is worthwhile revealing what this fact entails. The most general theory behind the procedure has been established by Sylvester (Burnside, Panton, 1960, Vol. II, p. 194) and amounts to putting the equation (1) in the form of a sum of two perfect cubes:

(2)

In this case the equation can be easily solved to give

(3)

where we denoted

(4)

i.e. w_j are the cubic roots of the unity:

(5)

The problem of solution of the cubic equation is thus translated into that of finding the quantities α₁, α₂, β₁, β₂ from equation (2) as functions of the coefficients a₀, a₁, a₂, a₃, which are usually related to physical situations. This can be done as follows: identifying the equations (1) and (2), gives the following system of equations

(6)

Now, we always may assume that α₁ and α₂ are the roots of a quadratic equation, which we write in the form:

(7)

This equation entails the natural identities

(8)

which we try to put in relation with the system of equations (6). For instance, we can obtain one equation by multiplying the first of the equations (8) by β₁ the second one by β₂ and then adding the results. The coefficients of b₀, b₁, b₂ in the new equation are given by a₀, a₁, a₂ from (6). Likewise, another equation may be obtained when multiplying the first of the equations (8) by β₁α₁, the second one by β₂α₂ and then add the resulting equations. The neat result for this procedure is the following system of equations for b₀, b₁, b₂:

(9)

This system has the solution defined, up to an arbitrary factor, by the equations

(10)

showing that α₁ and α₂ from (2) are the roots of a quadratic equation:

(11)

called Hessian associated to the cubic equation (1). Now we must find β₁ and β₂ from (2). For this we can use any pair from the four equations (6). The result is the same up to a factor. Because we actually need only the ratio β₂/β₁, we can however use another, more direct method, having the advantage to exhibit directly the algebraic nature of β₁ and β₂. Namely, denoting the cubic from equation (1) by f (x), and using the equation (2), we find

(12)

whence the ratio β₁ and β₂ is given by equation

(13)

Thus, the solution of a cubic equation can, indeed, be reduced to that of a quadratic equation. Here the quadratic in question is the Hessian of the cubic. In some other methods of solution we might have some other quadratic, but it will still be in close relationship with the Hessian.

We can read on the equations above the following important conclusion: if the Hessian of a given cubic is a perfect square, then the cubic has this Hessian as a factor. Indeed, for α₁ = α₂ = α in (12) we have f (α) = 0, showing that α is also a root of the cubic (1). What we need to show is that it is a double root for that equation. To prove this, we notice that

(14)

where we denoted

(15)

Now, putting in (15) x = α, we have

(16)

On the other hand, because the Hessian is a perfect square we can write

(17)

so that, up to an arbitrary factor Ψ(α) can be written as

(18)

Calculating the curly bracket in (18) it turns out to be identically zero. This shows that Ψ (x) also has α as a root and, consequently, (x – α)² is a factor of f(x). This completes the proof of the announced property of the cubic equation. Now, in cases where the Hessian is a perfect square β₁ and β₂ from (2) are completely arbitrary as the equation (12) shows. Summing up the discussion so far, in this case we have the following general theorem:

Theorem 1: If a cubic equation has a Hessian that is a perfect square, then such a cubic contains the Hessian as a factor.

At this point of the discourse it is important to introduce a distinguished quantity, playing a very important role in the theory of the cubic equations. This quantity is the discriminant of the Hessian of a cubic also called the discriminant of the cubic itself. It is

(19)

and deserves this name because, like in the case of the quadratic equation, it determines the nature of the roots of cubic equation. For instance, the previous theorem shows that a cubic of null discriminant has two equal roots, and these cannot be but real. In cases where D ¹ 0 we have for the Hessian two distinct roots and the ratio of β₁ and β₂ is uniquely determined by equation (13). On that very equation we read another important conclusion: β₁ and β₂ have the same algebraic nature as α₁ and α₂. Specifically, if α₁ and α₂ are real so are β₁ and β₂; if α₁ and α₂ are complex so are β₁ and β₂. This conclusion is important, because it shows that the nature of the roots of equation (1) is indeed decided by the discriminant given in (19), as already stated above. We have the general theorem:

Theorem 2. If the Hessian of a cubic equation has distinct roots then the cubic equation itself has distinct roots. There are two cases:

a) if the discriminant of the cubic is positive, then the Hessian has two real roots, and the cubic itself has one real and two complex roots.

b) if the discriminant of the cubic is negative, then the Hessian has complex roots, and the cubic itself has three real roots

The proof of this theorem is based on the form (3) of the roots of a cubic. Indeed, equation (3) shows that if α₁ and α₂ are real, which happens for Δ > 0, k is also real so that among the solutions of the cubic x₁ is real while x₂ and x₃ are complex conjugated to each other. On the contrary, if α₁ and α₂ are complex, which happens when Δ < 0, then k is complex unimodular and all x_j from (3) are real. This completes the proof.

A special issue of this theory is the case where the cubic (1) is a perfect cube. The matrix characterized by such a perfect cube is proportional with the 3×3 identity matrix. Then it has a unique root, α say, and by the relations between roots and coefficients we can write

(20)

Then we have for the coefficients of the Hessian:

(21)

Based on this, we can prove the following theorem:

Theorem 3. If a cubic is a perfect cube, then it has a null Hessian. Reciprocally, if a cubic has a null Hessian then it is a perfect cube.

Notice that we understand the term ‘null’ here as referring to the class of polynomials: ‘null Hessian’ means null second degree polynomial, i.e. a polynomial having null coefficients. What we proved before was the first part of the theorem. The second part comes out as follows: if the Hessian is a null quadratic, then the relations (21) are satisfied. From those equations we have

(22)

Let us denote by α, the common value of these ratios. Then we have

(23)

showing that the cubic is indeed a perfect cube. This completes the proof.

The previous theorem poses a more general question, namely: if we know the roots of a cubic equation what are the roots of its Hessian? More to the point: given the roots of a cubic determine its Hessian. This problem can be solved directly by calculating the coefficients from (11), using the relations between roots and coefficients. These are

(24)

where x₁, x₂, x₃ are the roots of (1). At this point it is useful to adopt a notation often helping us in shortening the calculations. Let ψ (x₁, x₂, x₃) be a rational expression of the three roots of the cubic (1). Then we denote by

(25)

the sum over all cyclic permutations of the numbers 1, 2, 3 in the expression ψ(x₁,x₂,x₃). For instance, in this notation the relations (24) can be written as

(26)

Then, by performing the appropriate calculations, we have the following expressions for the coefficients of the Hessian of a given cubic

(27)

These are the general relations between the coefficients of the Hessian and the roots of the corresponding cubic equation. In particular, the theorems 1 and 3 above follow directly from (27) in the respective special cases. Another point to emphasize here is that the relations (27) allow us to calculate the discriminant of a cubic as a function of the roots. Such an expression is important in that for many cases we can have it in this form and we must be able to recognize it. To calculate the discriminant we simply use its definition (19) and the relations (27) above. After a lengthy, but otherwise straightforward calculation, we find

(28)

Like (27), this expression contains all the previous information. In particular, we can read directly on (28) that a given cubic equation cannot have positive discriminant, but only in case where it has two complex roots, as shown before.

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