Fast interpolation and multiplication of unbalanced polynomials
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Bibliographic record
Abstract
We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced polynomials. Let <Formula format="inline"><TexMath><?TeX $f\in \mathbb {Z}[x]$?></TexMath><AltText>Math 1</AltText><File name="issac24-47-inline1" type="svg"/></Formula> be an unknown polynomial and s, D be bounds on its total bit-length and degree, our new interpolation algorithm returns f with high probability using <Formula format="inline"><TexMath><?TeX $\tilde{O}\!\left(s\log D\right)$?></TexMath><AltText>Math 2</AltText><File name="issac24-47-inline2" type="svg"/></Formula> bit operations and O(slog Dlog s) black box evaluation. For polynomial multiplication, assuming the bit-length s of the product is not given, our algorithm has an expected running time of <Formula format="inline"><TexMath><?TeX $\tilde{O}\!\left(s\log D\right)$?></TexMath><AltText>Math 3</AltText><File name="issac24-47-inline3" type="svg"/></Formula>, whereas previous methods for (resp.) dense or sparse arithmetic have at least <Formula format="inline"><TexMath><?TeX $\tilde{O}\!\left(sD\right)$?></TexMath><AltText>Math 4</AltText><File name="issac24-47-inline4" type="svg"/></Formula> or <Formula format="inline"><TexMath><?TeX $\tilde{O}\!\left(s^2\right)$?></TexMath><AltText>Math 5</AltText><File name="issac24-47-inline5" type="svg"/></Formula> bit complexity.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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