Simultaneously preperiodic points for families of polynomials in normal form
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Bibliographic record
Abstract
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than m greater-than 1"><mml:semantics><mml:mrow><mml:mi>d</mml:mi><mml:mo>></mml:mo><mml:mi>m</mml:mi><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">d>m>1</mml:annotation></mml:semantics></mml:math></inline-formula>be integers, let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 1 comma ellipsis comma c Subscript m plus 1 Baseline"><mml:semantics><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:annotation encoding="application/x-tex">c_1,\dots , c_{m+1}</mml:annotation></mml:semantics></mml:math></inline-formula>be distinct complex numbers, and let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold f left-parenthesis z right-parenthesis colon equals z Superscript d plus t 1 z Superscript m minus 1 plus t 2 z Superscript m minus 2 plus midline-horizontal-ellipsis plus t Subscript m minus 1 Baseline z plus t Subscript m"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">f</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>:=</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mi>d</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msup><mml:mi>z</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mi>z</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mi>z</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf {f}(z):=z^d+t_1z^{m-1}+t_2z^{m-2}+\cdots + t_{m-1}z+t_m</mml:annotation></mml:semantics></mml:math></inline-formula>be an<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"><mml:semantics><mml:mi>m</mml:mi><mml:annotation encoding="application/x-tex">m</mml:annotation></mml:semantics></mml:math></inline-formula>-parameter family of polynomials. We prove that the set of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"><mml:semantics><mml:mi>m</mml:mi><mml:annotation encoding="application/x-tex">m</mml:annotation></mml:semantics></mml:math></inline-formula>-tuples of parameters<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis t 1 comma ellipsis comma t Subscript m Baseline right-parenthesis element-of double-struck upper C Superscript m"><mml:semantics><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>∈</mml:mo><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mi>m</mml:mi></mml:msup></mml:mrow><mml:annotation encoding="application/x-tex">(t_1,\dots , t_m)\in \mathbb {C}^m</mml:annotation></mml:semantics></mml:math></inline-formula>with the property that each<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript i"><mml:semantics><mml:msub><mml:mi>c</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">c_i</mml:annotation></mml:semantics></mml:math></inline-formula>(for<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i equals 1 comma ellipsis comma m plus 1"><mml:semantics><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">i=1,\dots , m+1</mml:annotation></mml:semantics></mml:math></inline-formula>) is preperiodic under the action of the corresponding polynomial<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold f left-parenthesis z right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">f</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf {f}(z)</mml:annotation></mml:semantics></mml:math></inline-formula>is contained in finitely many hypersurfaces of the parameter space<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper A Superscript m"><mml:semantics><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">A</mml:mi></mml:mrow><mml:mi>m</mml:mi></mml:msup><mml:annotation encoding="application/x-tex">\mathbb {A}^m</mml:annotation></mml:semantics></mml:math></inline-formula>.
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.
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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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