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PreprintCC BY-NC-SA 4.0arXiv

Superpolynomially Large Support in Every Irreducible Factor of Lacunary Polynomials

  • Bhawesh Mishra
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Abstract

We show that there exist infinitely many polynomials F∈ℚ[x] with exactly m nonzero coefficients such that every irreducible factor of F over ℚ has exp (Ω(√(m/log m))) nonzero coefficients. This places multiplication in a sharply different category from powers, composition, and other algebraic operations for which reverse-sparsity principles are known. It also gives an unconditional, degree-free obstruction to sparse factorization, complementing positive factor-sparsity results that impose hypotheses on coefficient height, exponent positions, reciprocal structure, or degree bounds.

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Cite

BibTeXRIS
BibTeX
@misc{mishra2026superpolynomially,
  title = {{Superpolynomially Large Support in Every Irreducible Factor of Lacunary Polynomials}},
  author = {Bhawesh Mishra},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22286},
  eprint = {2609.22286},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22286},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Superpolynomially Large Support in Every Irreducible Factor of Lacunary Polynomials
AU  - Bhawesh Mishra
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.22286
UR  - https://arxiv.org/abs/2609.22286
AB  - We show that there exist infinitely many polynomials F∈ℚ[x] with exactly m nonzero coefficients such that every irreducible factor of F over ℚ has exp (Ω(√(m/log m))) nonzero coefficients. This places multiplication in a sharply different category from powers, composition, and other algebraic operations for which reverse-sparsity principles are known. It also gives an unconditional, degree-free obstruction to sparse factorization, complementing positive factor-sparsity results that impose hypotheses on coefficient height, exponent positions, reciprocal structure, or degree bounds.
ER  -