The formal proof constructs a complex polynomial p such that
p(0)=0,p′(0)=1,
and for every critical point c of p,
cp(c)>1.
Thus at z=0 there is no critical point satisfying
∣c∣∣p(0)−p(c)∣≤∣p′(0)∣=1,
which disproves Smale's conjectured universal constant K=1.
The counterexample has very large unspecified degree and violates the bound only by a smal…
The problem is open, and the repository says so: what is proved are criteria for it and two instances, not the general case.
Every compared statement that concludes Problem 10.61 does so for a quadratic setup - a real root α>1 of X2−aX−b whose conjugate has modulus below one - and both instances are quadratic. The arbitrary-degree material is conditional ingredients: for the family…
Let T be a complete first-order theory in discrete or continuous logic, let M≺U, and let μ∈Mx(U) be Borel-definable over M. The paper proves that the following three conditions are equivalent:
(i)μ is a frequency interpretation measure (fim) over M;
(ii)μ is definable over M and its canonical random extension rμ is generically stable over…
VibeMathed recorded: Mauri and Moraga's Question on Log Calabi-Yau Pairs with Big Boundary. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Ciliberto et al. Question on Effective Divisors of Positive Self-Intersection. Consult the linked registry entry for the original report and scope.
For n <= 2, we proof that the Generalized Vanishing Conjecture holds, and we have a failure at dimension 3 (including a counter-example).
This improves the original failure at dimension 5.
The exact Mul4 instance is resolved: its unrestricted XOR--AND multiplicative complexity is 9. The restricted bilinear/quadratic value 9 was classical; the new theorem proves that arbitrary nonlinear Boolean reuse cannot lower it. This is a natural positive special case of the Boyar--Find question, not a solution of the general (n,m) problem. The lower bound is a conceptual structural argument. A complete Lean 4 for…
VibeMathed recorded: Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication. Consult the linked registry entry for the original report and scope.
Fixed four-terminal planar DAG; positive route-cost differences for m≥2, with non-attained suprema; signed/zero only for m=1, whose attainment is unstated. No topology-wide/unrestricted-planar claim.