Sparse Arithmetic Zero Challenge
S4
smale derived
smale derived
External: externally open
τ response: further investigation
Given a univariate integer polynomial described by a short arithmetic construction, can the number or structure of its integer zeros be bounded in terms of the arithmetic complexity of constructing the polynomial?
Current τ response
See the paired Sparse Arithmetic Zero Challenge — Challenge Response on the Results lane for the program's current response status, registry evidence, verification route, and external-review boundary.
Current status: further investigation.
Challenge statement
Given a univariate integer polynomial described by a short arithmetic construction, can the number or structure of its integer zeros be bounded in terms of the arithmetic complexity of constructing the polynomial?
Why this challenge is in the ledger
Strong fit for τ at the interface of sparse representation, arithmetic complexity, integer zeros, factorization, and addressability.
τ-facing burden
Show whether τ supplies a genuine structural route to bounding integer zeros or only renames arithmetic-circuit complexity. Note: Smale’s tau-conjecture is unrelated to Panta Rhei’s τ except by notational coincidence.
First reviewer questions
- Does τ produce a nontrivial bound or only rename arithmetic-circuit size?
- Does the proposed route interact with known arithmetic-circuit lower-bound obstacles?
- Is there a bridge from τ-address complexity to standard sparse polynomial complexity?
Source anchors
Source anchors are background references, not endorsements of Panta Rhei claims.