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A Septic with 99 real Nodes

Oliver Labs
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Abstract

We find a surface of degree 7 in real projective three-space P^3(R) with 99 real nodes within a family of surfaces with dihedral symmetry: First, we consider this family over some small prime fields, which allows us to test all possible parameter sets using computer algebra. In this way we find some examples of 99-nodal surfaces over some of these finite fields. Then, the examination of the geometry of these surfaces allows us to determine the parameters of a 99-nodal septic in characteristic zero. This narrows the possibilities for μ(7), the maximum number of nodes on a septic, to: 99 <= μ(7) <= 104. When reducing our surface modulo 5, we even obtain a 100-nodal septic in P^3(F_5).

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Oliver Labs. 2004-09-20. A Septic with 99 real Nodes. https://arxiv.org/abs/math/0409348

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