Note on Lürotits Type of Plane Quartic
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چکیده
T=x+y+z+3xy+3xz+3xy+ 3xh+ £ « y " V + a^a£. Set y = Xz. Then Î becomes £ == x + r2œV + rgt?£ + • + r6s , r2=:3\ +A\-\-3, rz=D\ +E\ r^3\+F\*+M\+G\+3, r 5 = JBX + „5TX 3 + 2A + f7X, r6 = X 6 + cxX 5 -f . . . + c5X + 1. Now r = x zfc cc» — z, viz., is of type TT, if and only if (21) r a s r 4 s r 5 s 0 , rj s 1, r6 s 1 (mod 7) ; while T is a perfect cube if and only if (22) rB == r5 == 0, r4 == 5r , r6 == 6r 3 (mod 7). Since r2rs ~ 0 for every X, D = i? = 0. Hence (21) is excluded, so that (22) must hold for every X. We may therefore remove the term yz from T and proceed as in § 10. Or we may proceed with (22) and show that T=-(x + y + z-2Ayzy.
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