412 hitchcock. 



6. Application to the Problem of Expressing Polynomials as 

 Determinants whose Elements are Linear Polynomials. 



If we take M = 3 we have q = n(2, A) — 1 = K. We may then 

 regard K equations of type (37) as so many linear equations in the 

 A' + 1 unknowns Ci,- ■ -Cr^. The A-row determinants from the 

 matrix of the coefficients are polynomials of degree A; their elements 

 are linear in x; and, aside from a constant factor, they must be equal 

 to the respective C's which enter. We therefore have the means of 

 setting up polynomials of degree A in 3 variables in the form of 

 determinants with linear elements and vanishing at n — 1 given points. 



By use of the theory of matrices, Dickson has recently given a 

 highly elegant proof that all homogeneous polynomials in two and in 

 three variables, as well as quadrics and sufficiently general cubics in 

 four variables, can be expressed as determinants with linear elements , 

 and that no other general polynomials can be so expressed. 2 He has 

 shown how such a transformation can be accomplished for an arbi- 

 trary plane curve by employing no irrationalities except the roots of a 

 single equation of degree A. The present investigation is equally 

 general if we have no regard for rationality: a polynomial of degree A 

 through n — 1 arbitrary points is an arbitrary polynomial. With 

 regard to rationality, however, the method of the present article is less 

 general, for Dickson does not assume any points on the curve to be 

 known. But an explicit formulation of the polynomial as a function 

 of the n points ai, • • • a„_i and x is not without utility. 



To take a concrete case, let .V = 3 and A = 3. Let it be required 

 to express as a determinant with linear elements the cubic through 

 the nine points ai • • • a 9 . Let the six points ai • • • a 6 be regarded as 

 n(3, 2) points corresponding to the case N = 3, A = 2. Let AY • • A 6 

 be the respective quadrics which vanish at five of these six points, the 

 omitted point having corresponding subscript. Let {.vij) and (kij) 

 denote the determinants of the components of the vectors x, a,, a, and 

 &k, a,, a, respectively. We may chose bi = [xa,]; so that b,«ay = 

 ( xij ) . We may write aio for an arbitrary tenth point. Witli the nota- 

 tion already used Cio will be our required cubic. We may form six 

 equations of the form (37) each of which involves the cubics Ct • -Cio- 

 Three of these are sufficient and may be taken thus: 



(j-17)A 1 (a 7 )C 7 + (.rlS)A 1 (a 8 )t7 8 + (^19)£ 1 (a 9 )C 9 + Crlio)Ei(aio)C 10 =0 

 (.r27)A 2 (a 7 )C 7 + (.r28)A 2 (a 8 )C 8 + (.r29)A 2 (a 9 )C 9 + (.r2 10 )A 2 (a 1 o)C 1 o =0 

 (x37)E 3 (an)C 7 + (z38)£ 3 (a 8 )C 8 + (.r39)A 3 (a 9 )<7 9 + (.r3 ]0 )A 3 (a 1 o)C 1 o =0 



2 Trans. Amer. Math. Soc. Vol. 22, No. 2 (April 1921), pp. 167-179. 



