What does this research mean for the field?

The maximum number of distinct integer roots for constant-free straight-line programs of length 1 through 8 is exactly 1, 2, 3, 3, 4, 5, 5, and 6, respectively. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Determine the exact maximum number of distinct integer roots achievable by univariate integer polynomials computed with at most eight arithmetic gates under the Shub-Smale tau conjecture framework.

August 23, 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 eight

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Determine the exact maximum number of distinct integer roots achievable by univariate integer polynomials computed with at most eight arithmetic gates under the Shub-Smale tau conjecture framework.
Constructed the complete reachable computational state spaces through depth 7 (including 25,844,905 states at depth 6 and 1,048,460,912 states at depth 7) using exact integer arithmetic.
Implemented a last-gate lemma to exhaustively evaluate root counts one depth beyond the stored frontier without full frontier storage.
Cross-validated computational pipelines against Markstrom's census benchmarks and independent computer-algebra root counters across 284 sample polynomials.
Established the exact maximum distinct integer root ladder for gate lengths 1 through 8 as z_max(1..8) = 1, 2, 3, 3, 4, 5, 5, 6.
Determined that the minimal gate complexity to produce 4, 5, and 6 distinct integer roots is 5, 6, and 8 gates respectively, revealing growth plateaus at tau = 4 and tau = 7.
Demonstrated that polynomials constructed by iterating monic maps of degree ≥ 2 (such as x^2 - c) yield depth-independent integer root counts, limiting their ability to generate large root counts.

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Implication

Computational study establishes exact maximum integer root counts for arithmetic circuits up to eight gates, demonstrating growth plateaus in the Shub-Smale tau conjecture.

Cite This Study

Felipe Santibañez-Leal (2026) studied this question.

synapsesocial.com/papers/6a8aad977677a34114445c56 https://doi.org/10.5281/zenodo.22035884

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Implication

Computational study establishes exact maximum integer root counts for arithmetic circuits up to eight gates, demonstrating growth plateaus in the Shub-Smale tau conjecture.

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

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Implication

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 eight

Key Points

Discussion

Implication

Cite This Study

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