How many zero-sum subspaces exist for the multiplicative inverse function?

Counting subspaces where the inverse sum vanishes

Let q=2^n and let F_q be the finite field with q elements. Every subspace in the paper is an F_2-subspace, and all dimensions are taken over F_2. For a subspace E of F_q, the paper studies the sum S(E)=Σ_{x∈E\{0}} x^{−1}, where x^{−1} is the multiplicative inverse in F_q. The quantity N_{n,k} is the number of k-dimensional F_2-subspaces E with S(E)=0.

Carlet's conjecture, stated as Conjecture 1.1 in the paper, asserts that N_{n,k}>0 for every n≥6 and every 3≤k≤n−3. Equivalently, the multiplicative inverse function on F_{2^n} is not kth-order sum-free in that range. The paper's main theorem, Theorem 1.3, proves this conjecture.

A useful reduction is that only vector subspaces need to be checked. Carlet observed that if V is a vector subspace and a∉V, then Σ_{x∈a+V} x^{−1}=L'_V(a)/L_V(a)=b_0/L_V(a)≠0, where L_V is the subspace polynomial of V and b_0 is its coefficient of X. Affine subspaces that do not contain 0 therefore never give a zero sum, so the count N_{n,k} over vector subspaces is the right object.

Why the central range resisted earlier point counting

Earlier work made substantial progress. Carlet introduced the subspace-polynomial formulation, proved the symmetry N_{n,d}=N_{n,n−d}, and identified the next-to-leading coefficient τ(H) of a subspace polynomial as the key quantity. Carlet and Hou used Frobenius-invariant subspaces and point counting, proving the conjecture for even n and for relatively small dimensions. Ebeling, Hou, Rydell and Zhao found additional arithmetic families. Hou and Zhao proved absolute irreducibility of the symmetric polynomials F_k and Θ_k, and their point-counting estimates settled the conjecture for n≥8 in the ranges 3≤k≤⌊n/3⌋+2 and ⌈2n/3⌉−2≤k≤n−3, as well as every fixed dimension 3≤k≤12 when n≥k+3. A later paper handled every nonprime extension degree n.

The remaining cases were confined to prime extension degrees, and the existing methods did not give a uniform treatment when k grows proportionally with n. The obstruction is concrete: the two hypersurfaces involved have degrees deg F_k=2^k−2 and deg Θ_k=2^k−1. When k is proportional to n, these degrees are exponential in n, and the usual point-counting error terms do not provide uniform information in the central range. Nontrivial intermediate subfields used in many known constructions are also unavailable there.

Cheng's argument takes a different route and does not rely on the primality of n. It gives a uniform quantitative estimate for arbitrary n in the range n≥2k+1 while avoiding point counting on those high-degree hypersurfaces.

Tracking the whole coefficient distribution, not just its zero

For a d-dimensional subspace H, let L_H be its subspace polynomial and let τ(H) be its next-to-leading coefficient. Instead of tracking only the subspaces with τ(H)=0, the paper studies the entire distribution of τ(H) among d-dimensional subspaces that contain 1. The reason is structural: the count at τ(H)=0 alone does not satisfy a closed recursion when passing from dimension d to dimension d+1, whereas the full distribution does.

Concretely, for 1≤d≤n and b∈F_q, define h_d(b) as the number of d-dimensional subspaces H with 1∈H and τ(H)=b, set B_d to the Gaussian binomial coefficient counting the (d−1)-dimensional subspaces of the hyperplane, and let p_d(b)=h_d(b)/B_d. Then p_d is a probability distribution: if H is chosen uniformly from the d-dimensional subspaces containing 1, p_d(b)=P(τ(H)=b). For d=1 the only such subspace is F_2={0,1}, whose subspace polynomial is X^2+X, so p_1=δ_1.

Lemma 2.3 connects this distribution back to the target count: p_d(0)=N_{n,d}/G_{n,d}, where G_{n,d} is the Gaussian binomial coefficient counting the d-dimensional subspaces of F_q. The paper also records the duality N_{n,d}=N_{n,n−d} and the identity S(A)=τ(H)^2 relating a subspace to the orthogonal complement of the image of its subspace polynomial under the trace pairing.

A recursion whose two-step contraction handles growing dimensions

The algebraic engine is the map φ(x)=x^2+x. Its kernel is F_2 and its image is T=ker Tr. If H is a (d+1)-dimensional subspace containing 1, then φ(H) is a d-dimensional subspace of T, and conversely every d-dimensional subspace V of T has inverse image φ^{−1}(V) that is a (d+1)-dimensional subspace containing 1. This gives a bijection between the two families. Comparing the two highest terms of L_H=L_V∘φ yields τ(H)=1+τ(V).

Theorem 3.1 turns this into an exact recursion for the distributions: p_{d+1}(1)=p_d(0), and for b≠1, p_{d+1}(b)=[1−p_d(0)+(s−1)Σ_{v∈F_q^*} ψ((b+1)/v)p_d(v)]/(q−s), where s=2^d and ψ(x)=(−1)^{Tr(x)}. Writing this as p_{d+1}=T_{2^d} p_d defines a linear operator T_s on real-valued functions on F_q. The operator preserves the coordinate sum and fixes the uniform distribution u=q^{−1}1, so p_{d+1}−u=T_s(p_d−u).

The Fourier analysis of T_s is where the uniformity comes from. Lemma 4.1 shows that on the subspace H of functions with zero coordinate sum, ||T_s f||_2^2=α_s^2||f||_2^2+(1−α_s^2)|⟨f,w_0⟩|^2 with α_s=√q(s−1)/(q−s), and that T_s w_0=w_1. On w_0^⊥∩H the operator is α_s times an isometry onto w_1^⊥∩H. In each step all but one direction are contracted; the exceptional direction is mapped to a direction almost orthogonal to the exceptional direction of the next step, so two successive steps give a uniform contraction. This is what makes dimensions growing proportionally with n tractable.

The quantitative bound and the central-range asymptotic

Theorem 1.2 is the quantitative form of the result. Let k≥3, let n≥2k+1, and put r=⌊(k−1)/2⌋. Then |2^n N_{n,k}/G_{n,k} − 1| < 2^{r^2+r} 2^{n(1−r/2)}. In particular, if k≥10, then N_{n,k} > G_{n,k}/2^{n+1}.

The proof groups the k−1 transfers from f_1=p_1−u to f_k=p_k−u into r consecutive pairs. For the pair beginning at d=2j+1, the assumption n≥2k+1 gives q=2^n≥2^{2d+5}=32s^2, so the two-step estimate applies and ||f_{2j+3}||_2 < (2·2^{2j+1}/√q)||f_{2j+1}||_2. Iterating over j=0,…,r−1 and using Σ_{j=0}^{r−1}(2j+2)=r^2+r gives ||f_k||_2 < 2^{r^2+r} q^{−r/2}. Since p_k(0)=N_{n,k}/G_{n,k} and u(0)=1/q, any coordinate is bounded by the Euclidean norm, which yields the stated inequality after multiplying by q=2^n.

The strongest statement is the uniform central-range estimate: for all sufficiently large n, sup over n/3<k≤(n−1)/2 of |2^n N_{n,k}/G_{n,k} − 1| < 2^{−n^2/18+4n/3}. This says that throughout the central interval the proportion of zero-sum k-spaces is asymptotic to 2^{−n}, uniformly in k, and the endpoint k=(n−1)/2 is included. That is also the range where the degrees of the hypersurfaces are largest relative to the field size, which is exactly where the earlier point-counting approach lost uniformity.

From a quantitative estimate to Carlet's conjecture

Theorem 1.3 assembles the conjecture from several pieces. By the symmetry N_{n,d}=N_{n,n−d}, one may replace k by min(k,n−k) and assume k≤n/2. If n is even and k=n/2, the subfield F_{2^k}⊆F_{2^n} is a zero-sum k-space, because inversion permutes its nonzero elements and the sum of all elements of F_{2^k} is zero. This is the elementary middle-dimensional subfield construction.

In every remaining case n≥2k+1. For k≥10, Theorem 1.2 gives N_{n,k}>0. It remains only to consider 3≤k≤9, and Hou and Zhao proved the required conclusion for every 3≤k≤12 under the condition n≥k+3, which holds automatically here. Combining these cases gives N_{n,k}>0 throughout 3≤k≤n−3, proving the conjecture.

The paper also notes that the recursion of Theorem 3.1 provides an exact procedure for computing the numbers N_{n,k} when n is fixed: formula (3.4) can be evaluated by an additive Fourier transform after the nonzero field elements are permuted by inversion. The author states this computational observation is not needed for the proof.

What the bound does and does not claim

The result is a counting statement about F_2-subspaces of F_q with q=2^n, not about affine subspaces. The reduction to vector subspaces is justified by Carlet's observation that affine subspaces not containing 0 always give a nonzero inverse sum, so the count N_{n,k} is the relevant one.

The quantitative bound in Theorem 1.2 applies under k≥3 and n≥2k+1. The stronger uniform estimate (1.3) is stated for all sufficiently large n and for n/3<k≤(n−1)/2; it is not claimed for every n or for the full range 3≤k≤n−3. The existence statement for the remaining small dimensions 3≤k≤9 relies on Hou and Zhao's earlier result for 3≤k≤12 when n≥k+3, and the middle-dimensional case k=n/2 for even n relies on the subfield construction.

The asymptotic N_{n,k}∼2^{−n}G_{n,k} is uniform for n/3<k≤(n−1)/2 as n→∞. It describes the proportion of zero-sum k-spaces among all k-dimensional F_2-subspaces, not the absolute count, and it is an asymptotic statement rather than a finite-n identity. The paper's method avoids point counting on the high-degree hypersurfaces and does not depend on the primality of n, which is what allows a uniform treatment when k grows proportionally with n.

The priority this paper establishes

The decision-relevant comparison in this paper is between what point counting could deliver and what the distribution recursion delivers. Point counting settled the conjecture for n≥8 in the ranges 3≤k≤⌊n/3⌋+2 and ⌈2n/3⌉−2≤k≤n−3, and for every fixed dimension 3≤k≤12 when n≥k+3, but its error terms degrade when k is proportional to n because the hypersurface degrees 2^k−2 and 2^k−1 become exponential in n. Cheng's recursion instead contracts the deviation from the uniform distribution by a factor over each pair of dimensions, giving a uniform bound across the central interval n/3<k≤(n−1)/2.

For a reader deciding what to take from this paper, the sharpest takeaway is the central-range statement: the proportion of zero-sum k-spaces is asymptotic to 2^{−n}, uniformly in k, including the endpoint k=(n−1)/2. That uniformity, rather than the bare existence of one zero-sum subspace per dimension, is the paper's distinctive contribution, and it is what closes the prime-extension-degree cases that earlier methods left open.

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