Rigidity of Exact Contractivity in the Complex Polynomial Bohnenblust--Hille Inequality

@misc{pellegrino2026rigidity,
  title = {{Rigidity of Exact Contractivity in the Complex Polynomial Bohnenblust--Hille Inequality}},
  author = {Daniel M. Pellegrino and Eduardo V. Teixeira},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36887},
  eprint = {2610.36887},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36887},
}

TY  - UNPB
TI  - Rigidity of Exact Contractivity in the Complex Polynomial Bohnenblust--Hille Inequality
AU  - Daniel M. Pellegrino
AU  - Eduardo V. Teixeira
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.36887
UR  - https://arxiv.org/abs/2610.36887
AB  - The complex polynomial Bohnenblust--Hille inequality bounds the coefficient ℓ_2m/(m+1) norm of a complex m-homogeneous polynomial by its supremum norm, with a constant independent of the number of variables. For each fixed integer N≥3, we characterize the choices of N degree-m monomials for which the inequality holds with optimal constant $1$ for every polynomial supported on them, provided m exceeds an explicit threshold depending only on N. This holds precisely when their exponent vectors contain no nontrivial three-term arithmetic progression; equality for a nonzero polynomial then holds only for monomials. We also prove that, in the same degree range, every polynomial with at most N nonzero coefficients that violates the constant-one inequality has a unique coefficient of maximal modulus, whose exponent is the center of such a progression in its support. Exact contractivity also follows from a comparability condition on the two largest coefficients, with a degree threshold that necessarily depends on the number of nonzero monomials.
ER  -