MathDB

Linear independence of polynomial compositions conjecture

For every fixed integer m≥1m\ge 1 and every collection of pairwise distinct nonconstant polynomials f1,…,fm∈C[x]f_1,\ldots,f_m\in\mathbb{C}[x], there exists an integer d0d_0 such that, for every d≥d0d\ge d_0, a generic polynomial g∈C[x]g\in\mathbb{C}[x] of degree dd has the property that the polynomials g∘f1,…,g∘fmg\circ f_1,\ldots,g\circ f_m are linearly independent over C\mathbb{C}.

References

Primary source

arXiv

Additional references

algebradeep-learning-theorypolynomial-identities
Posted

Progress summary

Refreshed
Claimed progress

A new paper claims progress on the conjecture by proving several important special cases, but the full conjecture remains open.

The conjecture predicts when applying a generic polynomial to distinct polynomial inputs preserves linear independence, with consequences for identifying symmetries in deep networks. A public discussion restates this claim but records no proof, counterexample, or proposer and date.

August 27, 2026: several cases claimed established

The paper Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks reports proofs of several cases of a conjecture generalizing Newman–Slater’s theorem, including a variant for compositions passing through the origin. It presents applications to parameter identifiability in deep neural networks, while explicitly leaving the general conjecture open. These claims are not independently verified in the supplied record.

Current status (as of August 2026): The general conjecture remains open; several special cases and applications are claimed in the August 2026 paper, but those claims are unverified.

Sources

Solutions 0

No solutions have been posted yet.