Is the asymptotic Makar-Limanov rank conjecture true for noncommuting polynomials?

The theorem bounds normalized rank, not exact zeros

The paper proves the asymptotic Makar-Limanov rank conjecture. Fix an algebraically closed field k of characteristic zero and a nonconstant polynomial f in finitely many freely noncommuting variables. For each positive integer n, let r_f(n) be the minimum, over all tuples of n-by-n matrices over k, of the rank of the evaluated matrix f(Z_1, ..., Z_d). The theorem states that the infimum of r_f(n)/n over all n at least 1 is zero.

The denominator matters. The claim is not that some matrix value of f is exactly zero, and not that the rank is bounded by a fixed constant at every size. It says that for every real epsilon greater than zero there exist a size n and matrices Z_1, ..., Z_d in M_n(k) with rank f(Z_1, ..., Z_d) strictly less than epsilon times n. Positive rank at every finite size is allowed.

The paper notes that this establishes exactly the assertion recorded as Conjecture 6.1 by Volcic, attributed there to a 2017 talk at Bar-Ilan University. The authors also flag a separate question from the same source: whether r_f(n) is at most 1 whenever f is not constant on M_n(k)^d. That separate question is not settled by the theorem.

Exact matrix zeros can be impossible

The introduction explains why the conjecture replaces exact zeros with small normalized rank. For the polynomial 1 minus the commutator [x_1, x_2], the trace of I_n minus [X, Y] equals n, so no matrix value can be zero in characteristic zero. Exact matrix zeros therefore cannot play the same role as scalar zeros of commutative polynomials over an algebraically closed field.

The obstruction disappears when vanishing is measured by normalized rank. With X_n the sum over j from 1 to n-1 of j times E_{j,j+1} and Y_n the sum over j from 1 to n-1 of E_{j+1,j}, direct calculation gives I_n minus [X_n, Y_n] equal to n times E_{nn}. Its rank is 1, so its normalized rank is 1/n, which tends to zero even though an exact zero is impossible.

This example is the paper's stated motivation for the asymptotic formulation. It also clarifies what the theorem does and does not deliver: the infimum is zero, but the theorem makes no assertion about a finite attained zero or about a bound holding at every size.

Reduction to a two-variable polynomial with zero abelianization

The proof first reduces the general problem to a normalized two-variable form. Lemma 2 states that either f has a scalar zero, in which case r_f(1) is zero and the infimum is trivially zero, or there are a nonzero scalar c and a nonconstant polynomial h equal to c^{-1} f(xy, xy^2, ..., xy^d) in k<x, y>, written as h = 1 + g with g nonzero and with zero abelianization.

The substitution sending x_i to xy^i is injective, and every word in the support of g contains both letters. If D is the degree of h and A and M are the maximum numbers of x's and y's in such a word, then A and M lie between 1 and D-1. The reduction also gives a rank identity: for all n-by-n matrices X and Y, the rank of h(X, Y) equals the rank of f(XY, XY^2, ..., XY^d).

The paper emphasizes that this transfer of witnesses requires no surjectivity of the substitution on matrix tuples. The normalized polynomial h = 1 + g, together with the integers D, A, and M, is then fixed for the symbolic construction. Writing g as a sum of parts h_k grouped by the exact number k of y's, some h_k may vanish but h_M does not.

A two-derivation symbol algebra supplies finite-precision solutions

The core construction takes place in a symbol algebra S_p(F) of formal sums of a_r D^r over a lattice L_p equal to p^{-1} Z, with support bounded above. The algebra carries two commuting derivations delta and eta that vanish on k, a product star defined by a sum over j of Delta^j(U) delta^j(V) divided by j!, and an order function ord. Lemma 3 proves that this product is unital and associative, that k is central, and that ord(U star V) is at most ord U plus ord V.

Lemma 4 shows that for a rational c one can extend the coefficient field to a commuting differential field containing a nonzero e with delta e equal to zero and eta e equal to c e. On a lattice containing c, the symbol e D^{-c} is central and invertible, and multiplication by it for the star product is ordinary multiplication. This is what allows the construction to shift slopes without losing control.

Lemma 5 proves that for every nonzero f in k<x, y> the generic symbol f(D, v) is nonzero. Under the additional assumptions that every word of f contains x, contains exactly k_0 at least 1 copies of y, and contains at most A_0 copies of x, it gives the order lower bound ord f(D, v) at least 1 minus A_0 k_0. This bound is the quantitative input that later controls how much precision each correction step can achieve.

A factor-orbit estimate bounds denominators and forces termination

The abstract and introduction describe a factor-orbit estimate as the step that bounds the denominators of the correction exponents and proves finite termination at every prescribed precision. The introduction locates this step in Section 5, where an orbit estimate for irreducible factors under roots of unity controls those denominators.

This is the mechanism that turns an infinite symbolic construction into one that reaches any prescribed accuracy in finitely many steps. The paper's proof strategy is to construct solutions to increasing finite precision and then convert them into matrix values, so termination at each precision is what makes the later realization and descent possible.

The estimate is stated in the abstract as bounding denominators and proving finite termination at every prescribed precision. The evidence packet does not supply the numerical constants of the orbit estimate itself, so the reader should treat the bound as a structural guarantee rather than a specific numeric threshold.

Taylor and normal-order realization, then compression and descent

After the symbolic solutions are built, the paper realizes the symbols as infinite matrices with controlled support using a Taylor map and normal ordering. Finite compression then confines the error to boundary rows and columns, so that a finite-rank condition can be read off from the infinite-matrix picture.

The final step descends the resulting finite rank conditions to k. The proof of Theorem 1 assembles these pieces: for a positive integer T, Proposition 10 supplies a field F_T, a size p_T, and a symbol Z_T; Proposition 12 is applied with U equal to D, V equal to Z_T, H equal to h, R equal to A, and S equal to T. This yields matrices over F_T of size n_T equal to p_T T^2 at which the rank of h is at most q_T equal to 4 D A p_T T.

Setting r_T as the minimum of n_T and q_T, Lemma 13 then gives matrices X'_T and Y'_T over k of size n_T with rank of h(X'_T, Y'_T) at most r_T, hence at most 4 D A p_T T. The paper stresses that only this finite rank condition is descended; the differential field and its symbolic equations need not descend.

For 1 at most i at most d, the proof sets W_{i,T} equal to X'_T (Y'_T)^i. These are permitted matrices in the original minimum, and Lemma 2 gives 0 at most r_f(n_T)/n_T at most rank f(W_{1,T}, ..., W_{d,T})/n_T equal to rank h(X'_T, Y'_T)/n_T at most 4 D A / T. Since D and A depend only on the fixed input and its reduction, choosing T greater than 4 D A / epsilon makes the ratio less than epsilon for any positive epsilon.

The infimum is zero, but finite-size behavior stays open

The theorem is an asymptotic statement about the infimum over all positive sizes. It permits positive rank at every finite size, and it does not assert a finite attained zero or a bound at every size. The paper states this explicitly in the discussion following Theorem 1.

The paper also records a separate question from Volcic: whether r_f(n) is at most 1 whenever f is not constant on M_n(k)^d. The authors show that this separate question fails already at n equal to 3. They set p(x, y) equal to [x, [y, x][x, [y, x]^2]] and f(x, y) equal to 1 plus p(x, y). For arbitrary 3-by-3 matrices X and Y, writing P for p(X, Y), the trace of P and the trace of P^2 both vanish, and a symbolic computation implies P^3 equals det(P) times I_3. The eigenvalues of P are then c, c omega, and c omega^2 for some c in k with omega a primitive cube root of unity, so at most one eigenvalue equals minus 1 and the rank of I_3 plus P is at least 2.

Explicit matrices X_0 and Y_0 are given in the paper, and direct calculation yields p(X_0, Y_0) equal to a displayed 3-by-3 matrix whose rank added to I_3 is 2. Since p(0, 0) is zero, f is not constant on M_3(k)^2, which shows r_f(3) equals 2. This counterexample is to the separate finite-size question, not to the asymptotic conjecture, and the distinction is the paper's own.

A Lean 4 formalization accompanies the proof

The paper reports a Lean 4 formalization of Theorem 1 using Mathlib. The formalized statement quantifies over a field K that is algebraically closed and of characteristic zero, over a natural number d, and over every nonconstant free polynomial f in d variables, and concludes that the infimum of the set of real numbers r for which there exists a positive n with r equal to the minimum rank of f at n divided by n is zero.

The paper states that the main theorem uses only Lean's standard axioms propext, Classical.choice, and Quot.sound, with no sorry and no additional axioms. The authors also declare that the main results were obtained through interaction with an artificial intelligence agent system, MechMath Agent Team, and that GPT-6 Astra was used to assist in exploring proof strategies and revising the exposition, with the authors assuming full responsibility for all proofs, mathematical details, and correctness.

For a reader weighing the result, the formalization is a check on the proof as stated rather than an independent source of new mathematical content. The declaration of AI assistance is a disclosure about how the work was produced, not a limitation on the theorem's scope.

The priority is the asymptotic threshold, not a fixed rank bound

The paper's central contribution is a threshold statement: for every nonconstant free polynomial over an algebraically closed field of characteristic zero, the normalized rank can be pushed below any positive epsilon by choosing a sufficiently large matrix size. The proof's quantitative handle is the bound 4 D A / T, where D and A come from the normalized two-variable reduction and T is the precision parameter chosen larger than 4 D A / epsilon.

The most useful comparison in the paper is between this asymptotic threshold and the separate finite-size question. The theorem leaves open whether r_f(n) is at most 1 whenever f is not constant on M_n(k)^d, and the paper's own counterexample shows r_f(3) equals 2 for a specific f. A reader should therefore not read the asymptotic result as a uniform small-rank bound at each size.

The second comparison worth keeping is between exact zeros and normalized rank. The polynomial 1 minus [x_1, x_2] has no exact matrix zero, yet its normalized rank is 1/n at the displayed matrices. The theorem's value lies precisely in this replacement: it shows that the asymptotic vanishing phenomenon is general, while the finite-size behavior of r_f(n) remains a distinct and partly open question.

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