We compute the maximum number of rational points at which a homogeneous polynomial can vanish on a weighted projective space over a finite field, provided that the first weight is equal to one. This solves a conjecture by Aubry, Castryck, Ghorpade, Lachaud, O'Sullivan and Ram, which stated that a Serre-like bound holds with equality for weighted projective spaces when the first weight is one, and when considering polynomials whose degree is divisible by the least common multiple of the weights. We refine this conjecture by lifting the restriction on the degree and we prove it using footprint techniques, Delorme's reduction and Serre's classical bound.

翻译：我们计算了有限域上加权射影空间中齐次多项式所能达到的有理点零点最大数目，条件是第一个权重等于1。这一结果解决了Aubry、Castryck、Ghorpade、Lachaud、O'Sullivan和Ram提出的猜想：当第一个权重为1且多项式的次数能被各权重的最小公倍数整除时，对于加权射影空间存在某种类似Serre界的最优等式。通过放宽对次数的限制改进该猜想，并利用足迹技巧、Delorme约化以及Serre经典界完成了证明。