Every graph of maximum degree d admits a signing σ with ρ(A_σ) ≤ 2√(3(d-1)).
Hyperbolic polynomials and the Kadison-Singer problem
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abstract
Recently Marcus, Spielman and Srivastava gave a spectacular proof of a theorem which implies a positive solution to the Kadison-Singer problem via Weaver's $KS_r$ conjecture. We extend this theorem to the realm of hyperbolic polynomials and hyperbolicity cones, as well as to arbitrary ranks. We also sharpen the theorem by providing better bounds, which imply better bounds in Weaver's $KS_r$ conjecture for each $r>2$. For $r=2$ our bound agrees with Bownik et al.
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math.CO 1years
2026 1verdicts
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An Improved Upper Bound for the Bilu-Linial Conjecture via Interlacing Families
Every graph of maximum degree d admits a signing σ with ρ(A_σ) ≤ 2√(3(d-1)).