A $k$-Dimensional Version of the Largest Intersection Problem

Abstract: Suppose we have $r$ linearly independent hypersurfaces of degree $d$ in $\mathbb{P}<sup>m$ (or $\mathbb{A}<sup>m$) defined over a finite field $\mathbb{F}<em>q$, whose intersection is at most $k$-dimensional. What is the largest possible number of $\mathbb{F}_q$-rational points in the intersection? We conjecture an exact formula for this problem in both the projective and affine settings, assuming $q\geq d+1$. The case $k=m-1$ recovers the Beelen-Datta-Ghorpade conjecture [2] and the case $k=0$ recovers the zero-dimensional conjecture in [15]. Another interesting special case of the conjecture is $k=m-r$, which corresponds to the complete intersection of $r$ degree $d$ polynomials. The case $r=1$, $k=m-1$ was proven by Serre in [16] who showed that if $F$ is a degree $d$ homogeneous polynomial, then $$|V(F)(\mathbb{F}_q)|\leq dq^{m-1}+π</em>{m-2}(q)$$. We prove the case $r=2$ and $k=m-2$, that is, if $F_1, F_2$ are coprime, degree $d$ homogeneous polynomials, then $$|V(F_1,F_2)(\mathbb{F}<em>q)|\leq d^{2q^{m-2}+π}</em>{m-3}(q)$$.

• The paper formulates affine and projective k-dimensional conjectures for the maximum number of finite-field points in polynomial common zero sets of dimension at most k, recovering the Heijnen–Pellikaan and BDG frameworks as special cases.
• The paper constructs polynomial subspaces attaining the conjectured lower bounds and proves an EGH-conditional algebraic formulation using Artinian reductions, regular sequences, and lex-plus-powers methods.
• The paper proves that coprime degree-d forms satisfy |V(F₁,F₂)(𝔽q)| ≤ d²q^(m−2)+π_(m−3)(q), with equality characterized by a specific linear-factor configuration, and shows the k=m−2 case would imply BDG.

This paper by Yuxin Lin formulates and partially proves a $k$-dimensional generalization of the largest intersection problem for hypersurfaces over finite fields. The central objects are the quantities $e_r^{A}(d,m,k;q)$ and $e_r^{P}(d,m,k;q)$, defined as the maximum number of $\mathbb{F}_q$-rational points in the common zero set of $r$ linearly independent polynomials of degree $d$ in $\mathbb{A}^m$ (respectively degree-$d$ homogeneous polynomials in $\mathbb{P}^m$), subject to the constraint that the vanishing locus has dimension at most $k$. The paper proposes exact formulae for both quantities under the hypothesis $e_r^{A}(d,m,k;q)$0, unifies several prior conjectures as special cases, and proves the projective conjecture in the complete-intersection case $e_r^{A}(d,m,k;q)$1, $e_r^{A}(d,m,k;q)$2: if $e_r^{A}(d,m,k;q)$3 are coprime homogeneous forms of degree $e_r^{A}(d,m,k;q)$4, then $e_r^{A}(d,m,k;q)$5.

Background and motivation

The unconstrained problem—maximizing $e_r^{A}(d,m,k;q)$6 over $e_r^{A}(d,m,k;q)$7-dimensional subspaces $e_r^{A}(d,m,k;q)$8 of degree-$e_r^{A}(d,m,k;q)$9 forms—is equivalent to computing generalized Hamming weights of projective and affine Reed–Muller codes. The affine version was settled exactly by Heijnen and Pellikaan via the quantity $e_r^{P}(d,m,k;q)$0 built from lexicographically ordered exponent vectors [heijnen1998generalized]. Projectively, the Boguslavsky–Tsfasman conjecture fails for $e_r^{P}(d,m,k;q)$1 [datta2015conjecture], and the Beelen–Datta-Ghorpade (BDG) conjecture gives the corrected formula $e_r^{P}(d,m,k;q)$2, where $e_r^{P}(d,m,k;q)$3 is the smallest index with a nonzero coordinate of $e_r^{P}(d,m,k;q)$4 [beelen2022combinatorial]. Known cases of BDG include $e_r^{P}(d,m,k;q)$5 (Serre, Sørensen), $e_r^{P}(d,m,k;q)$6 (Boguslavsky), $e_r^{P}(d,m,k;q)$7 (Zanella), $e_r^{P}(d,m,k;q)$8, $e_r^{P}(d,m,k;q)$9, $\mathbb{F}_q$0 (Datta–Ghorpade), $\mathbb{F}_q$1, the top range of $\mathbb{F}_q$2, $\mathbb{F}_q$3 (Lin–Singhal), and all parameters for sufficiently large $\mathbb{F}_q$4 [singhal2025conjecturebeelendattaghorpade].

The $\mathbb{F}_q$5-dimensional conjectures

The paper introduces a combinatorial model based on the box

$\mathbb{F}_q$6

and the subset $\mathbb{F}_q$7 of elements of total degree at most $\mathbb{F}_q$8. After removing the $\mathbb{F}_q$9 "pure" elements $r$0 with $r$1 in one of the first $r$2 coordinates, the conjectured affine value $r$3 counts the lex-smallest elements of $r$4 up to the $r$5-th largest element of $r$6; it equals $r$7 when $r$8. The projective conjecture adds a term $r$9 with $d$0.

The framework recovers known results cleanly:

• $d$1: the affine conjecture becomes the Heijnen–Pellikaan theorem and the projective conjecture becomes BDG.
• $d$2: both reduce to the zero-dimensional conjecture of Lin–Singhal [lin2026largest].
• $d$3 (complete intersections): the affine statement asserts $d$4, proven by Lachaud–Rolland [lachaud2015number]; the projective statement asserts $d$5, achieved by Lachaud–Rolland's examples but previously proven only for $d$6 (Serre).

Two structural results support the framework. First, the conjectured values are genuine lower bounds: explicit constructions of subspaces $d$7 achieving $d$8 and $d$9 are given, using products of linear factors over chosen field elements and a direct-sum argument via leading monomials to verify $\mathbb{A}^m$0. Second, $\mathbb{A}^m$1 is non-decreasing in $\mathbb{A}^m$2 with other parameters fixed—a case analysis on the first nonzero index $\mathbb{A}^m$3 of $\mathbb{A}^m$4 that also plays a role in the reduction argument below.

An algebraic formulation via Artinian reductions

The paper connects the geometric conjectures to commutative algebra through Artinian reductions. For a zero-dimensional scheme $\mathbb{A}^m$5 disjoint from an $\mathbb{A}^m$6-hyperplane $\mathbb{A}^m$7, the reduction $\mathbb{A}^m$8 satisfies $\mathbb{A}^m$9 and $d$0. A key lemma shows that when the quotient has Krull dimension at most $d$1, one can extract a regular sequence of length $d$2 from any finite-dimensional subspace whose generated ideal cuts the dimension down accordingly; applying this to $d$3 and to the ideal of $d$4-points ($d$5) yields a regular sequence of degrees $d$6 in the defining ideal, or $d$7 after dehomogenization when a suitable rational hyperplane exists.

This motivates the Affine Algebraic Conjecture: for a standard graded Artinian algebra $d$8 whose ideal contains a regular sequence of degree $d$9, bounding $\mathbb{P}^m$0 should force $\mathbb{P}^m$1. The paper proves this conjecture follows from the Eisenbud–Green–Harris (EGH) conjecture. Under EGH, $\mathbb{P}^m$2 has a lex-plus-powers model, so its Hilbert function is encoded by a closed, compressed subset $\mathbb{P}^m$3. A compression lemma (via a Clements–Lindström-type exchange argument replacing the lex-max element of $\mathbb{P}^m$4 with the lex-min element outside it, never increasing the low-degree part) shows that among closed compressed sets of fixed size, the initial segments $\mathbb{P}^m$5 minimize the degree-$\mathbb{P}^m$6 part. Since $\mathbb{P}^m$7 achieves equality, the bound follows. The author notes plainly that Conjecture 2 does not imply the geometric conjectures without extra hypotheses: in the affine setting, $\mathbb{P}^m$8 does not guarantee a regular sequence of the required degrees, and in the projective case one needs a hyperplane meeting $\mathbb{P}^m$9 in at most $k$0 points.

Reduction: the $k$1 projective conjecture implies BDG

A notable structural result is that proving the projective conjecture at codimension $k$2 (for all intermediate degrees $k$3) would establish the full BDG conjecture. The proof handles an arbitrary $k$4 with $k$5 by splitting on $k$6:

• If $k$7, a coprime pair exists in $k$8 (using $k$9), so $e_r^{A}(d,m,k;q)$00, and monotonicity of $e_r^{A}(d,m,k;q)$01 in $e_r^{A}(d,m,k;q)$02 transfers the assumed bound.
• If $e_r^{A}(d,m,k;q)$03 has an $e_r^{A}(d,m,k;q)$04-linear factor, the linear-factor lemma plus the inequality $e_r^{A}(d,m,k;q)$05 give the result.
• If $e_r^{A}(d,m,k;q)$06 has no $e_r^{A}(d,m,k;q)$07-linear factor, Homma's bound $e_r^{A}(d,m,k;q)$08 for hypersurfaces without linear factors, combined with the assumed $e_r^{A}(d,m,k;q)$09 bound for $e_r^{A}(d,m,k;q)$10, yields the estimate; the key numerical step is $e_r^{A}(d,m,k;q)$11, which holds since $e_r^{A}(d,m,k;q)$12 for $e_r^{A}(d,m,k;q)$13.

The complete-intersection case $e_r^{A}(d,m,k;q)$14, $e_r^{A}(d,m,k;q)$15

The main theorem states that for coprime $e_r^{A}(d,m,k;q)$16 of degree $e_r^{A}(d,m,k;q)$17 in $e_r^{A}(d,m,k;q)$18,

$e_r^{A}(d,m,k;q)$19

with equality (for $e_r^{A}(d,m,k;q)$20) only when some element of $e_r^{A}(d,m,k;q)$21 has a linear factor ($e_r^{A}(d,m,k;q)$22) and no $e_r^{A}(d,m,k;q)$23-linear component appears with multiplicity $e_r^{A}(d,m,k;q)$24. The proof proceeds by induction on $e_r^{A}(d,m,k;q)$25, splitting into cases:

• $e_r^{A}(d,m,k;q)$26: writing $e_r^{A}(d,m,k;q)$27 reduces to Serre's bound on $e_r^{A}(d,m,k;q)$28 plus Lachaud–Rolland's affine degree estimate on the residual part.
• Multiplicity-one linear component (with $e_r^{A}(d,m,k;q)$29): using the linkage ideal $e_r^{A}(d,m,k;q)$30 for the residual scheme, the residual intersects the line $e_r^{A}(d,m,k;q)$31 in a degree-$e_r^{A}(d,m,k;q)$32 hypersurface, which bounds $e_r^{A}(d,m,k;q)$33 across the $e_r^{A}(d,m,k;q)$34 hyperplanes through $e_r^{A}(d,m,k;q)$35. Summing per-hyperplane estimates $e_r^{A}(d,m,k;q)$36 gives a strict deficit $e_r^{A}(d,m,k;q)$37 below the target.
• Linear component of multiplicity $e_r^{A}(d,m,k;q)$38: an induction on $e_r^{A}(d,m,k;q)$39 decomposing $e_r^{A}(d,m,k;q)$40 into linear components, components inside exceptional hyperplanes, and a remainder $e_r^{A}(d,m,k;q)$41, with two regimes depending on whether $e_r^{A}(d,m,k;q)$42 (the count of linear components plus exceptional hyperplanes) exceeds $e_r^{A}(d,m,k;q)$43. In the hard subcase where a point lies off all exceptional hyperplanes, a double-counting incidence argument over hyperplanes through that point, combined with the induction hypothesis, closes the gap.
• No $e_r^{A}(d,m,k;q)$44-linear component: again split on $e_r^{A}(d,m,k;q)$45; the point-off-hyperplanes subcase uses the floor-function refinement $e_r^{A}(d,m,k;q)$46 to gain the slack $e_r^{A}(d,m,k;q)$47.

In every non-extremal case the paper obtains strict improvements over the conjectured bound, confirming that equality forces the stated rigidity conditions.

Limitations and open questions

Several caveats are explicit in the paper. Both main conjectures are stated for $e_r^{A}(d,m,k;q)$48 and remain open in general; the proven case covers only $e_r^{A}(d,m,k;q)$49, $e_r^{A}(d,m,k;q)$50. The Affine Algebraic Conjecture depends on the EGH conjecture, which itself is known only for $e_r^{A}(d,m,k;q)$51, for $e_r^{A}(d,m,k;q)$52 with degree sequences $e_r^{A}(d,m,k;q)$53 or $e_r^{A}(d,m,k;q)$54, and for Gorenstein algebras in $e_r^{A}(d,m,k;q)$55; moreover, even a proof of the algebraic conjecture would not directly imply the affine geometric conjecture because the required regular sequence need not exist. The reduction from $e_r^{A}(d,m,k;q)$56 to BDG requires the $e_r^{A}(d,m,k;q)$57 conjecture at all intermediate degrees $e_r^{A}(d,m,k;q)$58, not just at $e_r^{A}(d,m,k;q)$59. Finally, the multiplicity-$e_r^{A}(d,m,k;q)$60 analysis must drop the $e_r^{A}(d,m,k;q)$61 assumption upon restriction to hyperplanes, which is why the inductive proposition is stated without it—an unavoidable feature of the method rather than a technical artifact.

Conclusion

The paper provides a coherent $e_r^{A}(d,m,k;q)$62-dimensional extension of the largest intersection problem, with conjectural exact formulae interpolating between the Heijnen–Pellikaan/BDG regime ($e_r^{A}(d,m,k;q)$63) and the zero-dimensional regime ($e_r^{A}(d,m,k;q)$64). Its contributions are threefold: lower-bound constructions showing the conjectured values are sharp candidates; an EGH-based algebraic reformulation with a complete proof conditional on EGH; and a full proof of the projective conjecture for complete intersections of two quadric-free-degree forms, together with a reduction showing that the $e_r^{A}(d,m,k;q)$65 case would imply the Beelen–Datta–Ghorpade conjecture. The remaining open question posed by the framework is whether the conjectures hold for general $e_r^{A}(d,m,k;q)$66, particularly whether the affine conjecture can be established independently of EGH.