Algebraic Forms - Orderly Production and Enumeration

ALGEBRAIC FORMS - ORDERLY PRODUCTION AND ENUMERATION This contravariant II is noteworthy as being strictly dual to f itself; i.e., if variables (u) and (x) are exchanged, and invariants and contravariants derived from the altered II, as from f, then the new II altered by the same exchange of variables would be a multiple of the old f. The discriminant of H contains as a factor the discriminant of f. This has led to the discovery of closed systems among the comitants of the system,—closed with respect to the action of an operator 6, the "Aronhold operation." Denote in actual coefficients the two cubics and connections are discovered among mixed forms of like orders.

For a ternary form of order four, the minimum number of covariant forms in a basis has not been determined; one author, Miss E. Noether, has progressed so far in approximations as to prove that it is above 300. For simultaneous systems of a small number of quadrics such work has been carried to a finish; but for higher orders it would be sport rather than science. Con nections are soonest found among forms of low degrees.

Irrational Invariants

are such as satisfy equations, algebraic, differential or functional, with invariant or covariant coefficients. Forms apolar to a given form satisfy an equation linear in their coefficients; e.g., the doubly apolar cubics f and II mentioned above, so Hilbert's form cp = irrationally covariant to a cubic satisfies an equation (bx) - a rela tion discussed in vol. i of the Transactions of the American Mathematical Society. "Binary quartics which are multiples of their own Hessian covariants are the squares of quadrics." Com binants and semicombinants furnish another illustration. A The second major problem is that of orderly production and enumeration. The internal structure of all invariant forms is known; and all that involve a given fundamental form or system, in a given degree, and one or more sets of variables to specified orders, can be written down mechanically. There will be some ambiguity, i.e., repetitions, owing to the zero forms. The third great problem concerning the totality of such invariants is com monly called Gordan's problem; from Paul Gordan who in 1869 was the first to solve it for binary forms. The question arises, is there a finite basis for the entire system ... , F a in terms of which all others can be represented linearly? If F is any in variant not in that basis, is there an identical equation with coefficients qi, q2, ... qi polynomials in the variables , Gordan's various proofs, that this basis must always exist, appeared first in 1869, later, revised, in 1871, and sub sequently. His proof was based on the nature of certain ele mentary operations which build up and knit together those in variant forms; more precisely, on the Gordan series developing a form in terms of polars and Cayleyan operations. This is given in somewhat improved form by Grace and Young (The Algebra of Invariants, As Jacques Deruyts has gone further, treating the enumeration problem for forms in any number of variables, and as the question of finite systems is treated with greater generality by Hilbert, the earlier methods need not be exhibited. Deruyts's method is generalized from that for binary forms used by Arthur Cayley, James J. Sylvester and Fabian Franklin; Hilbert's was entirely novel.

Semi-invariants as Sources of Invariant Forms.—Semi invariant had been defined by Cayley and Sylvester, for binary forms, as a polynomial in coefficients and variables of one or more fundamental forms, whose specific property it is to remain unaltered (identically) when the variables undergo the sub stitution xi=yi+Xy2, x2= y2 and the coefficients undergo the corresponding linear transforma tion, the variables (y) being replaced afterwards by (x) . Deruyts applies the same term to functions in more than two variables, unaltered by any substitution whose determinant lacks all con stituents below the principal diagonal. For quaternary forms such a substitution is this:— Like Alfredo Capelli, he considers for a quaternary system, co variants which may contain three sets of cogredient variables (xix2x3x4) (Yi. • .Y4), (zi . z4). Clebsch and Gordan had employed instead three unlike sets, two mutually contragredient, the other intermediate, its substitution like that of two-rowed determinants in either of the first two sets. If a covariant contains, when written in Clebsch-Aronhold symbols, •ir determinant factors in every term, and is of orders in' in the (x), tn2 in the (Y), tns in the (z), (where m2-' m3), its weights are called 'xi = , 72=r-1-m2, 7r3= , ir4= 7r, then it is shown that every primary covariant can be derived uniquely from a semi-invariant as source, which contains in every term 7r4 4-rowed symbolic determinant factors like (a b2c3(14) ; ir3-7r4 3-rowed determinant factors with scripts I, 2, 3, like (aib2c3); 7r2-- 7r3 2-rowed determinant factors with sub scripts 1, 2, like (alb2) ; ri —7-2 monomial factors like cti, etc.

The number of such sources that can be written for a given set of weights ri , , ir3 , 71-4 , may be denoted by , 77-2 , , ir-41 but is subject to reduction.

This precisely defined structure of a semi-invariant is the clue to the problem: how many semi-invariants of given weights , ir2 , ir3 , 7r4 are linearly independent? For it is possible to count, directly or by formulae, the number of different monomial expressions, rational and entire in the coefficients, which have the prescribed weights. As they are expressed in actual coeffi cients (not symbolic merely) they are linearly independent. Next, they are required to satisfy the three differential equations expressing invariance under the one-way substitutions (2 t), (32), (43), which change the weights respectively to (71-1- 1, T-2 — 1, , r4), y ir2+ 73- I, 74), (ri 7r2 7r3+ 1, I). These in volve as many conditions as there are terms of such weights; hence their number is to be subtracted from the first estimate. Then the overlapping conditions must be considered, etc. In all, the enumeration calls for as many items, additive and subtrac tive, as there are terms in a four-rowed determinant. This state ment is for forms homogeneous in four variables. Were the forms binary, only two weights and two types of factors in semi invariants would occur, and the results would be those of Syl vester and Franklin. For invariants (free from variables) of forms in any number of variables, the weights for the first dis tribution are all equal, and the calculation therefore slightly easier. But whatever the orders in variables, after the enumera tion for linearly independent comitants comes the reduction for those composed of rational factors, factors which are them selves covariants of the same or lower orders. This is the "tamisage" of Sylvester and Franklin.