Pencils of quadratic forms over finite fields (Q1827772)

The paper starts with a formula for the number \(N(\mathbf{P}=0)\) of common zeros of a non-degenerate pencil \textbf{P} of quadratic forms over a finite field \(F_q\) in terms of the discriminant if \(q\) is odd or Arf invariant if \(q\) is even. Then the authors concentrate on pencils spanned by so-called gap forms. Let \(K\) be a finite extension of the field \(\mathbf{F}_q\) of odd degree \(n=2m+1.\) For each \(1\leq i \leq n-1\) the \(i\)th gap form is the form \(Q_i(\alpha):=\text{tr}(\alpha\alpha^{q^i}).\) The main result says that if \(n=2m+1\) is a prime number, the \(\mathbf{F}_q\)-pencil \textbf{P} on \(K_0:=\ker(\text{tr})\) spanned by gap forms \(Q_{i_1},...,Q_{i_r}\) is non-degenerate and \(n>(q-1)^r\), then \(N(\mathbf{P}=0)\) equals \(q^{m-r}(q^m+q^r-1)\) if \(q\) is a quadratic residue mod\,\(n\) and \(q^{m-r}(q^m-q^r+1)\) otherwise. The authors provide the reader with examples and also give a combinatorial application of the results to counting binary strings with an even number of 1's prescribed distances apart.