ALGEBRAIC FORMS - FINITENESS OF SYSTEM OF INVARIANTS: HILBERT'S EXTENSIONS The third major problem is concerned with the finiteness of reduced systems of invariant comitants. Reduction by tamisage may show that, for degrees and orders proposed, all covariants are reducible, i.e., rationally expressible in terms of those of lower degrees. Beyond certain upper limits this must always occur. While Gordan and others proved this for binary forms, Hilbert established it for any finite number of forms in any finite number of variables. His widely inclusive principal theorem relates to algebraic forms. homogeneous, rational and integral in the elements of any number of independent systems. Now invariants, as here defined, are so constructed from one or more sets of variables strictly so called, and from the coefficients of one fundamental form or from those of several independent forms. The theorem states, "If a series of forms be given, rational and integral and homogeneous in a domain of a finite number of finite sets of elements (variables or parameters), then it is always possible to find a finite basis, or set of forms among those given, such that every other form in the given (finite or infinite) series shall be expressible linearly in terms of those basal forms, with coefficients rational and integral in the same domain." The proof is by mathematical induction, from forms in n variables to forms in n+ variables, with starting-point in the obvious truth of the statement when a single variable x is concerned. It proceeds upon the lemma that a linear function of linear functions of several variables is itself a linear function of those variables.
According to this theorem of Hilbert's, if the given series of invariants be denoted by Ai, A2, - arranged in ascending degrees in sets of parameters concerned, then among the A's is contained a basal set Bi, B2, . Bk, such that every A is expressible thus:— the M's being rational in the same domain as the given A's. But more than this is true: the M's, if not invariants, can be replaced by invariants, which in turn are either B's, or expressible in the B's just as the A's which include them. To turn the M's into invariants while leaving A's and B's unchanged is an opera tion most neatly carried out by Mertens' device, as follows (con fining the treatment to ternary forms, for brevity) :— Assume any linear transformation of the variables involved, say xi , x2 , x3 , and cogredient sets, with indeterminate para meters al , , (x3 , , 'yi , • • . , viz., Calculate the induced transformations, resulting from this, upon the coefficients of the one or more fundamental forms. Those will give the old set of coefficients rationally in terms of the new, with denominators some powers of the determinant (a10273) or A. Every M is thereby expressed as a form in the given domain and in the parameters (a), (0), (7); so also the B's, and these become merely powers of A multiplied into functions of quantities exclusive of the (a), ([3) and (-y). Now perform on every Ai, Bk and Mil, the operation P"i where 'IN is the weight of the invariant Ai . Every Ai and B reappears multiplied by a numerical constant (some power of 6) ; every Mik yields an invariant and a numerical factor, and the identity is undisturbed. Accordingly, by repetitions if necessary, every Ai is expressible in terms of products of B's with purely numerical coefficients.
Powers and products of powers of the B i constitute a subset among the Ai . The same argument applies to them, giving a second theorem due to Hilbert. There exist an unlimited series of identities among the products of powers of the basal invariants of a system, identically true when all are explicit in the original variables and coefficients of given forms. These identities are called syzygies, and when all significant terms are brought to one side of the identity, that side (as a function of the Bk) is a syzygant of the first kind among the invariants Bk. Among the syzygants of the first kind there exist then syzygants of the second kind, etc. But by a third theorem of Hilbert's the number of kinds of syzygants upon a given system of invariants is itself a finite number. Little has been done in concrete illustration of these theorems, or in the substitution of precise limits for the "finite" numbers involved; though Hilbert himself fixed an upper bound for the number of kinds of syzygies when m is the number of independent forms in the basis of the modulus or system, and worked out details for the system of three quadric surfaces defining a twisted cubic curve.