On the evaluation of some sparse polynomials
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Bibliographic record
Abstract
We give algorithms for the evaluation of sparse polynomials of the form <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P equals p 0 plus p 1 x plus p 2 x Superscript 4 Baseline plus midline-horizontal-ellipsis plus p Subscript upper N minus 1 Baseline x Superscript left-parenthesis upper N minus 1 right-parenthesis squared Baseline comma"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P=p_0 + p_1 x + p_2 x^4 + \cdots + p_{N-1} x^{(N-1)^2},</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> for various choices of coefficients <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript i"> <mml:semantics> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">p_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . First, we take <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript i Baseline equals p Superscript i"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">p_i=p^i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for some fixed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; in this case, we address the question of fast evaluation at a given point in the base ring, and we obtain a cost quasi-linear in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartRoot upper N EndRoot"> <mml:semantics> <mml:msqrt> <mml:mi>N</mml:mi> </mml:msqrt> <mml:annotation encoding="application/x-tex">\sqrt {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We present experimental results that show the good behavior of this algorithm in a floating-point context, for the computation of Jacobi theta functions. Next, we consider the case of arbitrary coefficients; for this problem, we study the question of multiple evaluation: we show that one can evaluate such a polynomial at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> values in the base ring in subquadratic time.
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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 | 0.003 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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) | 0.001 | 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