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The distribution of conjugates of algebraic integers

Speaker: Alexander Smith
Date of Talk: November 21, 2024
Upstream link: UCLA Colloquium

1. The Question

A totally positive algebraic integer (TPAI) is an algebraic integer \(\alpha \) whose conjugates are all positive real numbers. The TPAI with the smallest trace is \(1\), followed by \(\frac{3\pm\sqrt 5}{2}\) with a trace of \(1.5\). Next up is \(2\cos \left( \frac{2}{7}\pi \right)+2\). In fact, for any \(a\in \mathbb{Q} \), \(2\cos \left( a \pi \right)+2\) is a TPAI with trace strictly less than \(2\). This is an infinite family of TPAI’s.

The question is, what’s the threshold for which this happens? For which \(\Lambda \leq 2\) is it true that there are infinitely many TPAI’s with trace \(< \Lambda \), but only finitely many TPAI’s with trace \(< \lambda < \Lambda \)?

Here are some references for progress on this problem:

• Schur, 1917, \(\Lambda > \sqrt e\approx 1.65\ldots\).
• Siegel, 1945, \(\Lambda > 1.733\ldots\)
• Smyth, 1984, \(\Lambda > 1.773\ldots\). This established a computational method that brought a deluge of papers following it.
• Serre-Smyth, 2000. This showed that Smyth’s method was incapable of working for \(\Lambda \geq 1.89\ldots\).
• Smith, 2023. Turns out, Smyth’s method is perfect and “converges” to the right answer.
• Orloski-Sardari, 2023. Better yet, \(\Lambda < 1.815\ldots\)
• Orloski-Sardari-Smyth, 2024, \(\Lambda > 1.802\ldots\)

2. Measures!?

A cool idea first used by Schur was to define the measure \[\mu _ \alpha = \frac{1}{n} \sum _{j=1}^{n} \delta _{\alpha _j},\] where \(\alpha \) is a totally real algebraic integer and \(\alpha _1,\ldots, \alpha _n\) are its conjugates. Then, the trace of \(\alpha \) is just the trace of \(\mu _ \alpha \), given by \(\int x\ d \mu _ \alpha (x) \).

Schur then used the fact that the discriminant of the minimal polynomial of \(\alpha \) was always a nonzero integer to get the inequality \[\iint \log \left\lvert x - y \right\rvert \ d \mu _ \alpha (x) d \mu _ \alpha (y) \geq 0.\] I didn’t catch the rest of the argument, but somehow one argues that if one has good control over these quantities or “energies” for a specific sequence of algebraic integers, then one can get nontrivial absolute lower bounds on the traces of \(\mu _ \alpha \).

3. Narrative Structure

This was generally a really engaging talk to attend — it felt like there was a narrative thread from the statement of the trace problem through the progress and theoretical barries all the way up to the “resolution” by the speaker’s work.

I think this is a rare quality that a lot of mathematical writing (even expositional writing) struggles to capture. Smyth’s method actually working because it was correct in principle and “in the limit” was such a dramatic turn of events. It felt documentary-like in this way.