Algebraic Forms - General Laws of Structure

ALGEBRAIC FORMS - GENERAL LAWS OF STRUCTURE The first major problem in the theory of forms is the determina tion of general laws for structure of invariants. For binary forms containing one set of variables it is proved that invariant forms consist of a finite number of terms composed of symbolic factors like (ab) and (ax). For a ternary form structure of in variant forms, invariant comitants in one set of variables (x) and one set of contragredient variables (u), every invariant concomitant consists solely of terms containing symbolic factors of these types: (ux), (abc), (ax), (abu). Equivalent symbolic representations of any one invariant form differ by a form con taining in each term one or more of the above listed zero factors. Starting from more than one fundamental 'form, more types of symbolic factors would be admitted; so also if a fundamental form contains both kinds of variables, x's and u's.

Set of Three Quadrics.

Amongternary forms, after quadrics and systems of two quadrics, the next in point of simplicity and geometric interest is the set of three quadrics, or, what is closely related to such a set, the ternary cubic. If all invariants, co variants and contravariants are considered, together with mixed comitants containing both sets of variables, and then 34 distinct forms constitute a complete system for the ra tional expression of all others. The best known of these are, for the cubic f written symbolically f = etc.:— combinant of two forms of like order, (ax)n and (Ax)n is any simultaneous covariant of the two which is unchanged, save for a factor, when the first two are replaced by and and and M2 denoting any constants. One such is their Jacobian, and all others are com itants of that Jacobian, (Aa) . A semicombinant of two forms of different orders is unchanged when the form of higher order is replaced by a homogeneous combination; e.g., when binary and are replaced by and . These are conjectured to be useful in discussing intersections of plane curves, or of surfaces, of dif ferent orders.

Classification.—Degreeand order give a two-way arrange ment, order being the number of variable factors in every term of the homogeneous covariant; degree, the number of factors which are coefficients of the fundamental form. One problem is, to enumerate the different invariant forms of a specified order and degree, counting as one any two whose difference is a zero-form. This counting gives a result that increases rapidly with the degree. But invariants of lower degree can be combined, in rational integral aggregates, into such of higher degree. This indicates a reduction at each stage of the enumeration by discarding those rational and integral in lower stages. Will the final result be zero at and beyond some finite degree? It proves to be so for all forms of low order in the binary field, and for systems of two or three ternary quadrics. The third major problem is, then, that of the finiteness of reduced systems of invariant comitants.