Solving Nonlinear Polynomial Systems via Symbolic-Numeric Elimination Method
Bibliographic record
Abstract
Consider a general polynomial system S in x1, . . . , xn of degree q and its corresponding vector of monomials of degree less than or equal to q. The system can be written as M0 · [xq1, xq−1 1 x2, . . . , xn, x1, . . . , xn, 1] = [0, 0, . . . , 0, 0, . . . , 0, 0] (1) in terms of its coefficient matrix M0. Here and hereafter, [...] T means the transposition. Further, [ξ1, ξ2, . . . , ξn] is one of the solutions of the polynomial system, if and only if [ξ 1 , ξ q−1 1 ξ2, . . . , ξ 2 n, ξ1, . . . , ξn, 1] T (2) is a null vector of the coefficient matrix M0. Since the number of monomials is usually bigger than the number of polynomials, the dimension of the null space can be big. The aim of completion methods, such as ours and those based on Grobner bases and others [4, 5, 6, 7, 8, 10, 16, 18, 17, 12, 9, 20], is to include additional polynomials belonging to the ideal generated by S, to reduce the dimension to its minima. The bijection φ : xi ↔ ∂ ∂xi , 1 ≤ i ≤ n, (3) maps the system S to an equivalent system of linear homogeneous PDEs denoted by R. Jet space approaches are concerned with the study of the jet variety V (R) = {( u q , u q−1 , . . . , u 1 , u ) ∈ J : R ( u q , u q−1 , . . . , u 1 , u ) = 0 } , (4) where u j denotes the formal jet coordinates corresponding to derivatives of order exactly j. A single prolongation of a system R of order q consists of augmenting the system with all possible derivatives of its equations, so that the resulting augmented systems, denoted by DR, has order q + 1. Under the bijection φ, the equivalent operation for polynomial systems is to multiply by monomials, so that the resulting augmented system has degree q + 1. A single geometric projection is defined as E(R) := {( u q−1 , . . . , u 1 , u ) ∈ Jq−1 : ∃ u q , R ( u q , u q−1 , . . . , u 1 , u ) = 0 } . (5)
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".