By Betten D.
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Extra info for 4-Dimensional compact projective planes with a 5-dimensional nilradical
Topological planes, Adv. Math. 2, (1967), 1-60. 16. Salzmann, H. : Kollineationsgmppen kompakter vier-dimensionaler Ebenen, Math. Z. 117 (1970), 112-124. 17. Salzmann, H. : Kollineationsgruppen kompakter 4-dimensionaler Ebenen II, Math. Z. 121 (1971), 104-110. 18. Salzmann, H. : Elations in four-dimensional planes, TopologyAppL 3 (1973), 121-124. 19. Salzmann, H. : Compact Projective Planes, de Gmyter, Berlin, 1995.
This means that f'(t) = 0 can occur for only one t (y E ]R fixed) and this implies the strict monotonicity of f. It remains to check that f behaves correctly for Itl --+ o~. For t ~ oo the function f has the form f(t) = t3(3w+ - 4 z + ) + terms of lower order in t if 3w+ - 4z+ # 0 t(y 2 + ~1) + const. e. if (w+, z+) = ( - I , - ½ ) " For t ~ - c ~ one has similar expressions, replace the indices + by - . If 3w+ 4z+ # 0 then by the condition (R) this number is > 0. If 3w+ - 4z+ = 0 then the behavior for t ~ oo is determined by the term (y2 + ~ ) t which has the positive coefficient y2 + ~ .
D. BETTEN 288 This leads to the system of equations 3(1 + t)(3t - 1)w+ - 9t:w_ - t + ½ = 0 - 9 ( 1 + t):w+ + 3t(3t + 4 ) w _ + t + 3 = 0 with the unique solution w+ = w_ = I' Since Q(~, 1 , t ) = 0, we have proved O(t) _< Q ( w + , w _ , t ) < 0 for all t w i t h - 1 < t < 0, and therefore ~ ( t ) _< 0 for all t C I~. As a consequence, if(t) >__0 for all t C ~ . e. w+ = ~ , z + = , or w_ = ,z_ = , or both. In all cases we calculate f ' ( t ) = (y - (t + ½))2. This means that f'(t) = 0 can occur for only one t (y E ]R fixed) and this implies the strict monotonicity of f.
4-Dimensional compact projective planes with a 5-dimensional nilradical by Betten D.