Adversarial Wronskians: A proof of the Kung--Traub conjecture
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Matthew J. Colbrook · arXiv: 2610.40660v1 · Summary page
Why cover this story this week?
A speed limit for finding roots
Root-finding algorithms repeatedly query a function or its derivatives to home in on a zero. The Kung–Traub conjecture asks how fast that convergence can be when each iteration has a fixed evaluation budget and carries no memory between iterations. A new preprint proves the proposed upper bound, even when an algorithm chooses its queries adaptively and processes the answers without restrictions on its rules.
The proof turns this broad algorithmic freedom into an adversarial problem. Polynomial Wronskians—determinants built from functions and derivatives—supply a lower bound at a fixed scale. A diagonal construction then produces an entire function that defeats any claimed higher order of convergence. The authors also report formal verification in Lean. Beyond numerical analysis, this is a result about information: clever processing cannot extract more local accuracy than the permitted queries support.
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