Higher singularities of smooth functions are unnecessary by Igusa K.

By Igusa K.

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Dixmier and Zara [53] proved that there is no GQ of order (3,6). 1 it is easy to check that t e {4,6,8,11,12,16}. Nothing is known about £ = 11 or £ = 12. In the other cases unique examples are known, but the uniqueness question is settled only in the case t = 4. The proof of this uniqueness that appears in Payne and Thas [128] is that of Payne [110, 111], with a gap filled in by Tits. Chapter 2 Regularity, Antiregularity and 3-Regularity In this chapter the important notions of regularity, antiregularity and 3regularity are introduced, and the connections with planes, nets, Laguerre planes, inversive planes and subquadrangles are described.

Let z{ be any point of 7 7 \(T J - U T-1-1) and let U be the number of points of T1- u T-11- collinear with z^ We count in two ways the number of ordered pairs (z i; u), with Zi G P\(T- L U T^), ueT±U T±J-, and Zi - u, and obtain Y,U = 2{s + l)((s 2 - s)s + {s+ l)(s - 1)) = 2(s 3 - l)(s + 1). i Next we count in two ways the number of ordered triples (zi,u,u'), with Zi e ? \ ( r i U T i i ) , i i 1 u ' e r i U T i i , M ~ Zi ~u',u^ u', and obtain J2 U{U - 1) = 2(s + l)s(s 2 - s) + 2{s + l) 2 (s - 1) = 2(s 3 - l)(a + 1).

We have \V'\ = (s + l) 2 (s - 1) + 2(s + 1) = (s + l)(s 2 + 1). Let L be a line of B'. If L is incident with some point of U U U', then clearly L is of type uu', with u e U and w' G 17'. 1, L is again incident with s + 1 points of V'. 1, S' = (V, B', I') is a subquadrangle of order (s,f). Since IT") = (s + l)(st' + 1) we have t' = s, and so S' is a subquadrangle of order s. Since U UU' QV',\U\ = \U'\ = s + l, and each point of U is collinear with each point of U', we have {x,y}x' = U and fry}1-'1-'=U'.

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