On the complexity of the D5 principle
Bibliographic record
Abstract
The standard approach for computing with an algebraic number is through the data of its irreducible minimal polynomial over some base field k. However, in typical tasks such as polynomial system solving, involving many algebraic numbers of high degree, following this approach will require using probably costly factorization algorithms. Della Dora, Dicrescenzo and Duval introduced "dynamic evaluation" techniques (also termed "D5 principle") [3] as a means to compute with algebraic numbers, while avoiding factorization. Roughly speaking, this approach leads one to compute over direct products of field extensions of k , instead of only field extensions. In this work, we address complexity issues for basic operations in such structures. Precisely, let [EQUATION] be a family of polynomials, called a triangular set, such that k ← K = k[X1,...,Xn]/T is a direct product of field extensions. We write δ for the dimension of K over k, which we call the degree of T. Using fast polynomial multiplication and Newton iteration for power series inverse, it is a folklore result that for any ε > 0, the operations (+, X) in K can be performed in cnεδ1+ε operations in k, for some constant cε. Using a fast Euclidean algorithm, a similar result easily carries over to inversion, in the special case when K is a field. Our main results are similar estimates for the general case, where K is merely a product of fields. Following the D5 philosophy, meeting zero-divisors in the computation will lead to splitting the triangular set T into a family thereof, defining the same extension. Inversion is then replaced by quasi-inversion: a quasi-inverse [6] of α ∈ K is a splitting of T, such that α is either zero or invertible in each component, together with the data of the corresponding inverses.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.005 | 0.023 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.004 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.004 | 0.007 |
| Scholarly communication | 0.008 | 0.022 |
| Open science | 0.004 | 0.010 |
| Research integrity | 0.003 | 0.009 |
| Insufficient payload (model declined to judge) | 0.016 | 0.003 |
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".