Jacobi sums and new families of irreducible polynomials of Gaussian periods
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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="m greater-than 2"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m> 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta Subscript m"> <mml:semantics> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\zeta _m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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>-th primitive root of 1, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q identical-to 1"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">q\equiv 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> mod <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 m"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a prime number, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s equals s Subscript q"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">s=s_{q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a primitive root modulo <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f equals f Subscript q Baseline equals left-parenthesis q minus 1 right-parenthesis slash m"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f=f_{q}=(q-1)/m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We study the Jacobi sums <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Subscript a comma b Baseline equals minus sigma-summation Underscript k equals 2 Overscript q minus 1 Endscripts zeta Subscript m Superscript a ind Super Subscript s Superscript left-parenthesis k right-parenthesis plus b ind Super Subscript s Superscript left-parenthesis 1 minus k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:munderover> <mml:msubsup> <mml:mi>ζ</mml:mi> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mspace width="thinmathspace"/> <mml:mi>a</mml:mi> <mml:mspace width="thinmathspace"/> <mml:msub> <mml:mtext>ind</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mspace width="thinmathspace"/> <mml:msub> <mml:mtext>ind</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">J_{a,b}=-\sum _{k=2}^{q-1}\zeta _m ^{\, a\, \text {ind}_{s}(k)+b\, \text {ind}_{s}(1-k)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-equal-to a comma b less-than-or-equal-to m minus 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</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">0\leq a, b\leq m-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ind Subscript s Baseline left-parenthesis k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mtext>ind</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\text {ind}_{s}(k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the least nonnegative integer such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s Superscript ind Super Subscript s Superscript left-parenthesis k right-parenthesis Baseline identical-to k"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>s</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mspace width="thinmathspace"/> <mml:msub> <mml:mtext>ind</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>≡</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">s^{\, \text {ind}_{s}(k)}\equiv k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> mod <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript q Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P_{q}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q element-of script upper P"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">q\in \mathcal {P}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of irreducible polynomials of Gaussian periods, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta Subscript i Baseline equals sigma-summation Underscript j equals 0 Overscript f minus 1 Endscripts zeta Subscript q Superscript s Super Superscript i plus m j"> <mml:semantic
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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) | 0.001 | 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