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In [1], Branko Grunbaum asks: Is the configuration \(C_3(n, 1, 4)\) geometrically realizable for some \(n\)? In this talk, we apply Hensel's Lifting Lemma to Singer polynomials corresponding to the difference sets defining the configurations to prove that there exist infinitely many \( C_3(n,1, 4)\) and \(C_4(n, 1, 4, 6)\)configurations over the real plane. In the case of \(C_3(n)\)\)s, the blocks of three points are the usual straight lines and in \(C_4(n)\)s the blocks of four points are circles. While the former uses the classical group law on cubics, the latter employs the group law on conics to realize the configurations, (for example, see [3]). This is a part of ongoing research done in collaboration with Jeffrey Lanyon.

References:

1. Grünbaum, Branko., Configurations of Points and Lines, Graduate Studies in Mathematics vol 103, AMS, 2009.
2. Grünbaum, Branko., Which \((n_4)\)s exist? Geombinatorics, 9 (2000) 164 169–169.
3. Franz Lemmermeyer, Conics, a poor man's elliptic curves, pdf.