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ABSTRACT: Ever since the classical Greek dilemma of cube duplication, the cubic equation fascinated the human intellect. The host of geometrical and analytical problems it carries within nourished the intellect in growing by expanding the number of its creations. We can say that cubic equation still exercises fascination today with new uncoverings. We take it here as being the primary expression of three-dimensionality of the Space of our habitation. It characterizes measurable things, and has a differential geometry attached to it leading to conservation laws related to these things. In the long run, one can say that the cubic equation is related to the fundamental structure of the World in Einstein’s characterization. The essay presents algebraical and geometrical details.

I. Introduction

Cubic equation is the basis for constitutive laws as we accept them today, inasmuch as it allows us to algebraically characterize a 3×3 matrix, regardless if it is a tensor or not. The following treatment refers to the most general form of cubic equation having real coefficients. These coefficients are orthogonal invariants in the case of a 3×3 matrix which is tensor with respect to orthogonal group of Space, and therefore they bear physical meanings. We accept an extension of this physical meanings regardless of the tensor character of the matrix. For the present work, the basic source of information regarding the topics related to equations is the classical treatise of Burnside and Panton (Burnside, Panton, 1960). This treatise shall be understood as everywhere cited throughout the present work. We write the cubic equation, for convenience, in the so-called binomial form as

and assume that the coefficients a_k are real, displaying by a₀ the possibility of adjusting them by an arbitrary factor on account of the known arbitrariness allowed by the relations between the roots and the coefficients of an algebraic equation.

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