The Slope Polynomial and Collinear Points in Permutations
نویسنده
چکیده
We consider a polynomial – which we term the “slope polynomial” – that encodes information about slopes of lines defined by a point-set in finite affine planes. When the aforementioned point-set is the graph of a permutation, we show that constancy of the polynomial is equivalent to the permutation being linear. This has immediate consequences for the structure of the 3-uniform hypergraph of collinear triples of the graph of a permutation, an issue that is itself connected with important questions in harmonic analysis and combinatorial number theory.
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