Degree-three relations among quadratic forms: the value β₂(7)=16, and a conjectured family past n=12

Abstract

Let K be a field of characteristic zero and let q₁,…,qₘ be quadratic forms in n variables over K. In his work on uniqueness of powers-of-forms decompositions, Taveira Blomenhofer writes β₂(n) for the largest m for which some such family admits no nonzero algebraic relation of degree at most three; the only bound he names is β₂(n) ≤ C(n+1, 2), coming from linear relations, and he conjectures the lower bound β₂(n) ≥ ⌈ (n+2)(n+1)/6⌉, witnessed by the explicit family qᵢⱼₖ=(Xᵢ+Xⱼ+Xₖ)² over the index triples i ≤ j ≤ k with i+j+k ≡ 0 (mod n), offering as evidence a computer verification for n ∈ {7,…,12}. We determine β₂(n) exactly for 2 ≤ n ≤ 7. Counting the target space gives β₂(n) ≤ mmax(n):=max{m:C(m+2, 3) ≤ C(n+5, 6)} ∼ n²/120^(1/3), sharper than the stated bound by a factor tending to 2.466…; and we exhibit mmax(n) explicit squares of linear forms with one-digit integer coefficients whose binom(mmax(n)+2, 3) three-fold products are linearly independent, for n=4,5,6,7. The resulting values β₂(n)=7,9,13 for n=4,5,6 agree with what can be read off Table 1 of Friedman–Sturmfels–Wiesmann; β₂(7)=16 lies past that table, and past the 12 that the conjectured family supplies, so that family is not optimal from n=4 on. We also give what appears to be the first formal certificate for a case of the conjecture, at n=7, where 364 products of 12 quadrics are shown independent over every field of characteristic zero; and we correct a remark of the source paper, whose asserted generator degree kⁿ for the relation ideal of n+1 general k-forms in n variables has the exponent one too large — a degree count predicts kⁿ⁻¹, which is what we measure at seven pairs (n,k). All independence statements rest on exhaustive finite computations, identified where they occur, and are machine-checked in Lean 4 with no sorry. Outside the formal development we record, as labelled computations, that the conjecture also holds at n=13,14,15,16 — n=13 being the first case its source leaves open — and that β₂(n)=mmax(n) persists for n=8,…,11.

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