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III. An External Factor

As it can be seen from the above development the Hessian of a cubic is a key tool in constructing the roots of that cubic. Sometimes, in practical problems, the physical principles even allow us to know the Hessian before we know the cubic itself, and in such cases we need to figure out the corresponding cubic, more specifically to construct the roots of this cubic. The previous development shows that, given the roots of the Hessian only, one cannot know the corresponding cubic equation without ambiguity. In fact the equations (2) and (3) show that to a given Hessian corresponds a one-parameter family of triplets of numbers, each one of these triplets representing a given cubic equation. This indeterminacy is independent from the known property of indeterminacy allowed by the relations between roots and coefficients. As a matter of fact, it is even deeper than the equation (2) shows it, in the sense that the ratio k, which by equations (4) and (13) depends only on the quantities related to the cubic equation, may hide in itself an external phase completely independent of the cubic equation (This observation and the algebraic proof that follows are due to Dan Barbilian). In order to see the nature of this problem, we use the following identity between cubic itself (f), its Hessian (H) and its Jacobian (T)(see APPENDIX):

The expression in right hand side of this equation can be decomposed into two factors each of the third degree, for the cubic and its Jacobian are prime with respect to each other. On the other hand, the left hand side is a product of two perfect cubes, for the Hessian is a quadratic polynomial. The identity (29) then shows that each factor of the right hand side is proportional to a factor of the expression from the left hand side and this proportionality can be taken in two ways at will. However, for a fixed choice between those two ways, the proportionality factors are reciprocal to one another. Indeed, the Hessian can be factorized in infinite many ways as

where U and V are first degree binomials and m is any nonzero number. Thus the identity (29) can be written as the system

Adding these expressions, gives the result (2) only in a slightly different form, showing clearly where the external arbitrariness comes into play:

One can further decompose the right hand side into linear factors, to the effect that (32) becomes

This form allows us to find the roots of the cubic equation f = 0 in the form given by equation (3) with k ≡ m^-2. In case the roots are all real, k must be complex unimodular as before. For the sake of completeness, we mention that the Jacobian of a cubic can be also obtained from (31) as a difference of cubes:

This shows that the roots of Jacobian are of the same nature as the roots of the cubic itself. In formula (3) we do not have to change but the sign in both the denominator and numerator in order to get the roots of the corresponding Jacobian (see APPENDIX). This discussion also shows that the form (3) of the roots of a cubic equation is valid independently of the nature of the roots.

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