A coefficient normalization for positive definite forms

The result

A basis that bounds every coefficient.

Let p be a real homogeneous polynomial of positive even degree d, positive away from zero. Its symmetric multilinear polarization A satisfies A(x, …, x) = p(x).

There is a basis z₁, …, z_m such that

for every mixed tuple. Pure tuples have value one. This gives the absolute-value bound asked in Problem 11, printed page 3182, of the 2019 Oberwolfach report.

Check the result

A short proof, with exact checks.

The universal statement follows from the analytic proof. A separate derivation uses positive definite bilinear slices, Cauchy–Schwarz and a finite maximum argument.

The Python companion checks 12 rational examples, covering 120 coefficient classes, with 110 independent polarization cross-checks. It also checks the quadratic identity in degrees 2–24 and rejects two deliberately invalid certificates.

python3 verification/verify.py

Python 3.9+; no extra dependencies. These finite checks are not a formal proof of the universal theorem.

• Three-page paperPDF · complete proof
• Source & verification packageLaTeX, scripts, certificates and audits
• Fresh preprint-readiness reviewsThree independent rounds and documented dispositions
• Verification reportAssumptions, proof map and boundary checks
• Independent Cauchy–Schwarz proofA stronger local cone statement
• Priority auditSources, queries, findings and limits
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Context & attribution

An answer to the stated question.

Brandes's 2015 paper discusses pseudo-diagonal forms and proves the quadratic case. The 2019 problem asks about general positive definite forms after a change of variables.

The literature audit found no earlier full resolution. It cannot rule out unpublished or differently formulated antecedents.

Generative AI supplied the initial candidate and assisted the independent internal checks, research and writing.