Smallest poles of Igusa's and topological zeta functions and solutions of polynomial congruences
Bibliographic record
Abstract
Igusa's p-adic zeta function is associated to a polynomial f in several variables over the integers and to a prime p. It is a meromorphic function which encodes for every i the number of solutions M_i of f=0 modulo p^i. The intensive study of Igusa's p-adic zeta function by using an embedded resolution of f led to the introduction of the topological zeta function. This geometric invariant of the zero locus of a polynomial f in several variables over the complex numbers was introduced in the early nineties by Denef and Loeser. It is a rational function which they obtained as a limit of Igusa's p-adic zeta functions and which is defined by using an embedded resolution. I have studied the smallest poles of the topological zeta function and the smallest real parts of the poles of Igusa's p-adic zeta function. For n=2 and n=3, I obtained results by using an embedded resolution of singularities. I discovered that the smallest real part of a pole of Igusa's p-adic zeta function is related with the divisibility of the M_i by powers of p. I obtained a general theorem on the divisibility of the M_i by powers of p, which I used to obtain the optimal lower bound for the real part of a pole of Igusa's p-adic zeta function in arbitrary dimension n. I obtained this lower bound also for the topological zeta function by taking the limit.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".