Linear maps preserving matrices annihilated by a fixed polynomial
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Bibliographic record
Abstract
Let M n ( F ) be the algebra of n × n matrices over an arbitrary field F . We consider linear maps Φ : M n ( F ) → M r ( F ) preserving matrices annihilated by a fixed polynomial f ( x ) = ( x − a 1 ) ⋯ ( x − a m ) with m ≥ 2 distinct zeroes a 1 , a 2 , … , a m ∈ F ; namely, f ( Φ ( A ) ) = 0 whenever f ( A ) = 0 . Suppose that f ( 0 ) = 0 , and the zero set Z ( f ) = { a 1 , … , a m } is not an additive group. Then Φ assumes the form (†) A ↦ S ( A ⊗ D 1 A t ⊗ D 2 0 s ) S − 1 , for some invertible matrix S ∈ M r ( F ) , invertible diagonal matrices D 1 ∈ M p ( F ) and D 2 ∈ M q ( F ) , where s = r − n p − n q ≥ 0 . The diagonal entries λ in D 1 and D 2 , as well as 0 in the zero matrix 0 s , are zero multipliers of f ( x ) in the sense that λ Z ( f ) ⊆ Z ( f ) . In general, assume that Z ( f ) − a 1 is not an additive group. If Φ ( I n ) commutes with Φ ( A ) for all A ∈ M n ( F ) , or if f ( x ) has a unique zero multiplier λ = 1 , then Φ assumes the form (†) . The above assertions follow from the special case when f ( x ) = x ( x − 1 ) = x 2 − x , for which the problem reduces to the study of linear idempotent preservers. It is shown that a linear map Φ : M n ( F ) → M r ( F ) sending disjoint rank one idempotents to disjoint idempotents always assume the above form (†) with D 1 = I p and D 2 = I q , unless M n ( F ) = M 2 ( Z 2 ) .
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it