Borcea’s 2-variance conjecture
The supplied source identifies Borcea's -variance conjecture as the case of a conjecture concerning the distance between the critical points of a polynomial and those of its derivative. However, neither the source abstract nor the supplied background states the conjectured inequality or defines the variance and distance quantities precisely, so a complete quantified formal statement cannot be given without introducing unsupported notation.
References
Primary source
Additional references
- Borcea's 2-variance conjecture — arXiv — Teng Zhang
Progress summary
A 2026 preprint claims to prove Borcea’s conjecture for and supplies Lean formalizations, but the broader proof has not received independent mathematical assessment.
Borcea’s -variance conjecture concerns the distance between the critical points of a polynomial and those of its derivative. The general case was formulated as open in the foundational 2010 work.
Known results
- The conjecture holds for polynomials with at most three distinct zeros, including degree- polynomials (Borcea and collaborators, 2010).
- For polynomials with real zeros, the stronger bound was established (2010).
- The general problem was reduced to a matrix-theoretic conjecture and remained open (2010).
October 2026 claimed proof
Teng Zhang’s preprint claims a proof for and provides machine-checked Lean formalizations of the main results. This is a claimed resolution, not an independently assessed one; the formal checking covers the encoded statements and proofs, not necessarily the paper’s wider exposition.
Current status (as of October 2026): The conjecture is claimed proved and formally verified in Lean, but that claim remains unverified by independent mathematical assessment.
Solutions 0
No solutions have been posted yet.