Quadratic symmetric polynomials and an analogue of the Davenport constant (Q6097390)

In this intriguing paper, the authors define polynomial Davenport constants and give some of their properties over finite fields with prime cardinality. Given a multivariate polynomial \(F\), e.g. \(F=y_1^2+\lambda y_2\), and a length \(r\), the authors propose to replace each variable \(y_i\) by the symmetric sum \(x_1^i + x_2^i+\cdots + x_r^i\). This enables them to associate to each finite sequence \(S=a_1,a_2,\cdots, a_r\), a value \(\varphi_F(S)\) which is our example is \((x_1+\cdots+x_r)^2+\lambda(x_1^2+\cdots+x_r^2)\). The Davenport \(\varphi_F\)-constant \(D(\varphi_F,p)\) is the smallest length \(\ell\) such that any sequence of elements in \(\mathbb{F}_p\) having at least \(\ell\) elements contains a subsequence \(S\) such that \(\varphi_F(S)=0\). The case \(F=y_1\) corresponds to the usual Davenport case. This paper proceeds by giving bounds for \(D(\varphi_F,p)\) when \(F = a y_1^2+ b y_2 + c y_1\).