waring_polynomial_problem

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Waring problem for polynomials: Every homogeneous polynomial of degree k in d variables is a sum of k-th powers of linear forms. The minimum number of summands needed (the Waring rank) is not known in general.

Why this node was retired

The posted statement is

import Mathlib

theorem waring_polynomial_problem (k d : ℕ) (hk : 2 ≤ k) (hd : 1 ≤ d) :
    ∃ (s : ℕ), ∀ (f : MvPolynomial (Fin d) ℝ),
      f.totalDegree = k →
      ∃ (forms : Fin s → MvPolynomial (Fin d) ℝ),
        (∀ i, (forms i).totalDegree = 1) ∧
        f = ∑ i, (forms i) ^ k := by
  sorry

Real sums of even powers cannot represent negative forms; total degree alone also omits homogeneity.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Require homogeneous degree-k polynomials and allow real scalar coefficients ∑c_i*L_i^k, or work over an algebraically closed field of characteristic0 for an unweighted power formulation.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.