Let \(S_{N}\) denote the subset of \(F_{q}[t]\) containing all polynomials of degree strictly less than \(N\), and \(Y=(a_{i,j})_{R\times S}\in F_{q}\).Suppose that \(Y\) satisfies the following two conditions.
C1. \(a_{i,1}+\cdots+a_{i,S}=0\) \((1\leq i\leq R)\).
C2. \(Y\) has \(L\) columns with \(L\geq R\) such that: any \(R\) of these \(L\) columns are linearly independent, after removing any \(L-R+1\) of these \(L\) columns from \(Y\), we can find two disjoint sets of \(R\) linearly independent columns among the remaining \(S-L+R-1\) columns, and we assume that these \(L\) columns are the first \(L\) columns of \(Y\).
Consider the system of equations \[ a_{i,1}x_{1}+\cdots+a_{i,S}x_{S}=0\qquad 1\leq i\leq R.\tag{1.1} \] Let \(D_{Y}(S_{N})\) denote the maximal cardinality of a set \(A\subseteq S_{N}\) for which the equations in (1.1) are never satisfied simultaneously by distinct elements \(x_{1},\dots,x_{S}\in A\).
In this paper, the authors employ a variant of the Hardy-Littlewood circle method for \(F_{q}[t]\) to prove the following result.
Theorem. Assume that \(Y\) satisfies C1 and C2. There exists an effectively computable constant \(C=C(Y)>0\) such that for \(N\in\mathbb{N}\), \[ D_{Y}(S_{N})\leq q^{N}(\frac{C}{N})^{(L-R+1)/R}. \]