About two decades ago, Tsfasman and Boguslavsky conjectured a formula for themaximum number of common zeros that $r$ linearly independent homogeneouspolynomials of degree $d$ in $m+1$ variables with coefficients in a finitefield with $q$ elements can have in the corresponding $m$-dimensionalprojective space. Recently, it has been shown by Datta and Ghorpade that thisconjecture is valid if $r$ is at most $m+1$ and can be invalid otherwise.Moreover a new conjecture was proposed for many values of $r$ beyond $m+1$. Inthis paper, we prove that this new conjecture holds true for several values of$r$. In particular, this settles the new conjecture completely when $d=3$. Ourresult also includes the positive result of Datta and Ghorpade as a specialcase. Further, we determine the maximum number of zeros in certain cases notcovered by the earlier conjectures and results, namely, the case of $d=q-1$ andof $d=q$. All these results are directly applicable to the determination of themaximum number of points on sections of Veronese varieties by linearsubvarieties of a fixed dimension, and also the determination of generalizedHamming weights of projective Reed-Muller codes.
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机译：大约在二十年前，Tsfasman和Boguslavsky猜想出一个最大零点的公式，即$ r $线性独立的齐次多项式在$ m + 1 $变量中具有$ q $元素的有限域中系数为$ d $的线性独立齐次多项式可以在相应的$ m $维投影空间。最近，达塔（Datta）和古尔帕德（Ghorpade）证明，如果$ r $最多为$ m + 1 $，则该猜想是有效的，否则可能是无效的。 $。在本文中，我们证明了这个新的猜想对于$ r $的多个值成立。特别是，这在$ d = 3 $时完全解决了新的猜想。我们的结果还包括Datta和Ghorpade作为特例的积极结果。此外，在某些情况下，我们确定了零的最大数目，这些情况是较早的猜想和结果未发现的，即$ d = q-1 $和$ d = q $的情况。所有这些结果都可直接用于通过固定维数的线性子变量确定Veronese品种截面上的最大点数，以及确定投射Reed-Muller码的广义汉明权重。
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