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The polynomial degree of the Grassmannian \({\mathcal G}_{1,n,2}\). (English) Zbl 1172.51304

Summary: For a subset \(\psi\) of \(\text{PG}(N,2)\) a known result states that \(\psi\) has polynomial degree \(\leq r\), \(r \leq N\), if and only if \(\psi\) intersects every \(r\)-flat of \(\text{PG}(N,2)\) in an odd number of points. Certain refinements of this result are considered, and are then applied in the case when \(\psi\) is the Grassmannian \({\mathcal G}_{1,n,2} \subset \text{PG}(N,2)\), \(N = \binom{n+1}2 - 1\), to show that for \(n<8\) the polynomial degree of \({\mathcal G}_{1,n,2}\) is \(\binom n2 - 1\).

MSC:

51E20 Combinatorial structures in finite projective spaces
05C90 Applications of graph theory

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References:

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