Stepanov (circa 1970) created the polynomial method to limit the rational points of an algebraic curve over $\mathbb{F}_q$, leading to one of several alternative proofs of Weil's Riemann hypothesis for one-dimensional function fields. The essence of Stepanov's method is succinctly demonstrated on a genus one curve in section 1.6 of Baker and Wustholz's Logarithmic Forms in Diophantine Geometry (emphasizing its affinity with the methodology of transcendence theory), and the general case is given by Bombieri in the Bourbaki Seminar and exposed in Tao's blog.
Deligne's higher dimensional theorem on the purity of Frobenius eigenvalues on cohomology does not in general admit an interpretation by limiting the rational points of an algebraic variety over $\mathbb{F}_q$; the Lang-Weil bound $|X(\mathbb{F}_q) - q^{\dim{X}}| = O_X(q^{\dim{X}-1/2})$ (as $q$ varies) is in general sharp, due to the presence of $(2\dim{X}-1)$-dimensional cohomology, and lies no deeper than the one-dimensional case, from which it was originally derived.
But suppose $X \subset \mathbb{P}_{/\mathbb{F}_q}^N$ is a smooth complete intersection. In that case, Theoreme 8.1 in Deligne's "Weil I" yields $$ |X(\mathbb{F}_q)| \leq |\mathbb{P}^{\dim{X}}(\mathbb{F}_q)| + O(q^{{\dim{X}}/2}), $$ with an implied constant only depending on $\deg{X}$. (It is in fact an equality, but let me only ask about limiting the $\mathbb{F}_q$-points since this is what Stepanov's method does.)
The question. Has Stepanov's polynomial method (or another "elementary approach") been attempted in such a higher dimensional situation?
