What does this research mean for the field?

The foundational mathematical theories underlying Smale's 4th Problem, including the Shub-Smale and Koiran tau-conjectures, can be comprehensively synthesized and their basic polynomial root properties formally verified using the Lean 4 interactive theorem prover. Novelty: ClaimNovelty.SYNTHESIS. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Investigate whether the number of integer zeros of a polynomial computed by a short straight-line program is polynomially bounded by circuit length, resolving Smale's fourth problem.

August 30, 2026Open Access

On the Number of Integer Zeros of Straight-Line Polynomials and Smale's 4th Problem

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CNCharles EDOU NZE

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Implication

Mathematical analysis uncovers bounds on polynomial integer zeros from arithmetic circuits, highlighting structural connections between algebraic complexity classes and formal Lean 4 verification.

Key Points

Investigate whether the number of integer zeros of a polynomial computed by a short straight-line program is polynomially bounded by circuit length, resolving Smale's fourth problem.
Synthesized Descartes' rule of signs, Khovanskii's fewnomial theory, and Lenstra's rational root bounds for sparse polynomials within a directed acyclic graph framework.
Analyzed structural reductions connecting integer and real root bounds of arithmetic circuits to the algebraic complexity separations P_C ≠ NP_C and VP ≠ VNP.
Implemented machine-checked formalizations of foundational algebraic degree bounds and root uniqueness theorems using the Lean 4 interactive theorem prover.
Provided complete non-elliptical proofs bounding positive real roots by sign variations via Rolle induction, generalizing to Lenstra's O(t^2 log t) sparse bound.
Confirmed that polynomial bounds on integer zeros imply P_C ≠ NP_C under the BSS model and that real root bounds on sparse polynomial sums imply VP ≠ VNP.
Formally verified monomial and linear root uniqueness alongside power-of-two degree bounds in Lean 4 with 0 axioms, 0 linter warnings, and 0 sorry placeholders.

Cite This Study

Charles EDOU NZE (2026) studied this question.

synapsesocial.com/papers/6a93f0a36c1a8fb52e79cf4d https://doi.org/10.5281/zenodo.22148643

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Implication

Mathematical analysis uncovers bounds on polynomial integer zeros from arithmetic circuits, highlighting structural connections between algebraic complexity classes and formal Lean 4 verification.

On the Number of Integer Zeros of Straight-Line Polynomials and Smale's 4th Problem

Discussion

Member takes

Implication

Key Points

Cite This Study

On the Number of Integer Zeros of Straight-Line Polynomials and Smale's 4th Problem

Discussion

Member takes

Implication

Key Points

Cite This Study

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