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The aim is to provide an exact enumeration of distinct integer roots for integer polynomials based on gate complexity, particularly testing the limits of the Shub-Smale tau conjecture.

August 4, 2026Open Access

The bottom of the Shub-Smale tau conjecture: an exact census of integer roots for constant-free straight-line programs of length at most seven

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FSFelipe Santibañez-LealOpen University of Cyprus

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Preprint details an exact count of integer roots for polynomials in programs with limited gates, suggesting implications for complexity theory.

Key Points

The aim is to provide an exact enumeration of distinct integer roots for integer polynomials based on gate complexity, particularly testing the limits of the Shub-Smale tau conjecture.
Exhaustive computation of all reachable states for polynomial evaluation up to depth 7.
Cross-validation with a computer-algebra root counter on 284 polynomials and comparison to prior exhaustive census.
Implementation of elementary stall theorems for detailed analysis of root counts across polynomial families.
Exact counts of distinct integer roots were established, showing z_max(1..7) = 1, 2, 3, 3, 4, 5, 5.
The minimum cost for achieving 4 distinct roots is 5 gates and for 6 roots is exactly 8 gates.
Found plateaus in growth function and confirmed z_max(8) >= 6.

Cite This Study

Felipe Santibañez-Leal (2026) studied this question.

synapsesocial.com/papers/6a71993fc47350ef8e494bee https://doi.org/10.5281/zenodo.21763182

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Implication

Preprint details an exact count of integer roots for polynomials in programs with limited gates, suggesting implications for complexity theory.

The bottom of the Shub-Smale tau conjecture: an exact census of integer roots for constant-free straight-line programs of length at most seven

Discussion

Member takes

Implication

Key Points

Cite This Study

The bottom of the Shub-Smale tau conjecture: an exact census of integer roots for constant-free straight-line programs of length at most seven

Discussion

Member takes

Implication

Key Points

Cite This Study

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