Hadamard products and binomial ideals
Bibliographic record
Abstract
We study the Hadamard product of two varieties V and W, with particular attention to the situation when one or both of V and W is a binomial variety. The main result of this paper shows that when V and W are both binomial varieties, and the binomials that define V and W have the same binomial exponents, then the defining equations of V⋆W can be computed explicitly and directly from the defining equations of V and W. This result recovers known results about Hadamard products of binomial hypersurfaces and toric varieties. Moreover, as an application of our main result, we describe a relationship between the Hadamard product of the toric ideal IG of a graph G and the toric ideal IH of a subgraph H of G. We also derive results about algebraic invariants of Hadamard products: assuming V and W are binomial with the same exponents, we show that deg(V⋆W)=deg(V)=deg(W) and dim(V⋆W)=dim(V)=dim(W). Finally, given any (not necessarily binomial) projective variety V and a point p∈Pn∖V(x0x1⋯xn), subject to some additional minor hypotheses, we find an explicit binomial variety that describes all the points q that satisfy p⋆V=q⋆V.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
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".