Step roots of Littlewood polynomials and the extrema of functions in the Takagi class
Bibliographic record
Abstract
Abstract We give a new approach to characterising and computing the set of global maximisers and minimisers of the functions in the Takagi class and, in particular, of the Takagi–Landsberg functions. The latter form a family of fractal functions $f_\alpha:[0,1]\to{\mathbb R}$ parameterised by $\alpha\in(-2,2)$ . We show that $f_\alpha$ has a unique maximiser in $[0,1/2]$ if and only if there does not exist a Littlewood polynomial that has $\alpha$ as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi–Landsberg functions with $\alpha\in(-2,1/2]\cup(1,2)$ . For $(1/2,1]$ , we show that the step roots are dense in that interval. If $\alpha\in (1/2,1]$ is a step root, then the set of maximisers of $f_\alpha$ is an explicitly given perfect set with Hausdorff dimension $1/(n+1)$ , where n is the degree of the minimal Littlewood polynomial that has $\alpha$ as its step root. In the same way, we determine explicitly the minima of all Takagi–Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to $[-2,-1/2]\cup[1/2,2]$ .
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How this classification was reachedexpand
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 | 0.004 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".