A note on nonsingular forms in a finite field
L. Carlitz · Proceedings of the American Mathematical Society · 1956
such that (2) det (QiQi + * + WrQr) #? 0 for arbitrary . ., Ir) 5= (0, , 0), tsGGF(q). This result suggests the possibility of finding r homogeneous forms fi, * * *, fr of degree m, where (q, m) = 1 such thatf1, ,fr vanish simultaneously only at (0, , 0), and secondly (3) f = W1fi + + rfr (i E GF(q)) has no singular point (except at (0, , 0)) for arbitrary (i not all zero. To construct such forms let : be a number of GF(qr) such that fl 013 * * *, 1 are linearly independent relative to GF(q); Hensel first proved the existence of such d. Now put