Generic polynomial composition linear independence

Prove that for each integer k≥2 there exists an integer R(k) such that, for every r≥R(k), a nonempty Zariski open subset of polynomials of degree at most r consists of activations σ for which σ(p₁),…,σ(p_k) are linearly independent for every collection of k nonconstant pairwise distinct multivariate polynomials p₁,…,p_k.

Background

The paper identifies linear independence of polynomial compositions as the algebraic problem underlying identifiability of multilayer perceptrons with generic polynomial activations. The conjecture seeks to generalize the Newman–Slater theorem, which establishes linear independence for sufficiently high powers p₁r,…,p_kr, from monomial activations to generic polynomial activations.

The authors prove the conjecture for two polynomials and in several degree-dependent settings, but leave open the degree-independent assertion in which the threshold R depends only on the number k of input polynomials and the good activation set is Zariski open. Its origin-passing variant would imply identifiability results for general deep polynomial MLPs without bias vectors.

References

We conjecture that not only monomial activations behave like that, but almost all polynomial activations:

Let k \geq 2. There is an integer R(k) such that, for every r \geq R(k), there is a nonempty Zariski open subset U_{k,r} \subseteq K[z]{\leq r} with the following property: For every \sigma \in U{k,r} and all multivariate polynomials p_1, \ldots, p_k that are nonconstant and pairwise distinct, the polynomials \sigma(p_1), \ldots, \sigma(p_k) are linearly independent.

— Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks  (2608.27113 - Kohn et al., 27 Aug 2026) in Conjecture 1, Section 1 (Introduction)

We wish to point out that it would also be interesting to prove the following weaker version of Conjecture~\ref{conj:strong}, where the bound R only depends on the number k of polynomials, but the Zariski open subset U is allowed to depend on their maximum degree m:

Let k \geq 2. There is an integer R(k) such that, for every r \geq R(k) and every m \geq 1, there is a nonempty Zariski open subset U_{k,r,m} \subseteq K[z]{\leq r} with the following property: For every \sigma \in U{k,r,m} and all p_1, \ldots, p_k \in K[t] of degree at most m that are nonconstant and pairwise distinct, the polynomials \sigma(p_1), \ldots, \sigma(p_k) are linearly independent.

— Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks  (2608.27113 - Kohn et al., 27 Aug 2026) in Conjecture 2 (weak), Section 2.3, after Theorem 2.5

Let k \geq 2. There is an integer R(k) such that, for every r \geq R(k), there exists a nonempty Zariski open subset U_{k,r} \subseteq K[z]0_{\leq r} with the following property: For every \sigma \in U_{k,r} and every collection of nonzero, pairwise distinct p_1,\dots,p_k \in K[t]0, the polynomials \sigma(p_1),\dots,\sigma(p_k) are linearly independent.

— Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks  (2608.27113 - Kohn et al., 27 Aug 2026) in Conjecture 3 (origin-passing), Section 3

Let k \geq 2. There is an integer R(k) such that, for every r \geq R(k) and every m \geq 1, there is a nonempty Zariski open subset U_{k,r,m} \subseteq K[z]{\leq r}0 with the following property: For every \sigma \in U{k,r,m} and all p_1, \ldots, p_k \in K[t]0_{\leq m} that are nonzero and pairwise distinct, the polynomials \sigma(p_1), \ldots, \sigma(p_k) are linearly independent.

— Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks  (2608.27113 - Kohn et al., 27 Aug 2026) in Conjecture 4 (weak, origin-passing), Section 3