6b. Sums of roots

[math] \newcommand{\mathds}{\mathbb}[/math]

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

Let us discuss now a generalization of the Butson obstruction from Proposition 6.4, which has been our main source of obstructions, so far. Let us start with:

Definition

A cycle is a full sum of roots of unity, possibly rotated by a scalar,

[[math]] C=q\sum_{k=1}^lw^k\quad,\quad w=e^{2\pi i/l}\quad,\quad q\in\mathbb T [[/math]]

and taken in a formal sense. A sum of cycles is a formal sum of cycles.

The actual sum of a cycle, or of a sum of cycles, is of course 0. This is why the word “formal” is there, for reminding us that we are working with formal sums. As an example, here is a sum of cycles, with [math]w=e^{2\pi i/6}[/math], and with [math]|q|=1[/math]:

[[math]] 1+w^2+w^4+qw+qw^4=0 [[/math]]

We know from Proposition 6.3 above that any vanishing sum of [math]l[/math]-th roots of unity must be a sum of cycles, at least when [math]l=p^a[/math] is a prime power. However, this is not the case in general, the simplest counterexample being as follows, with [math]w=e^{2\pi i/30}[/math]:

[[math]] w^5+w^6+w^{12}+w^{18}+w^{24}+w^{25}=0 [[/math]]

Indeed, this sum is obviously not a sum a cycles. However, this sum vanishes indeed, as shown by the following computation:

[[math]] \begin{eqnarray*} w^5+w^6+w^{12}+w^{18}+w^{24}+w^{25} &=&w^5+w^{15}+w^{25}\\ &+&w^0+w^6+w^{12}+w^{18}+w^{24}\\ &-&w^0-w^{15}\\ &=&0+0-0\\ &=&0 \end{eqnarray*} [[/math]]

The following deep result on the subject is due to Lam and Leung :

Theorem

Let [math]l=p_1^{a_1}\ldots p_k^{a_k}[/math], and assume that [math]\lambda_i\in\mathbb Z_l[/math] satisfy:

[[math]] \lambda_1+\ldots+\lambda_N=0 [[/math]]

• [math]\sum\lambda_i[/math] is a sum of cycles, with [math]\mathbb Z[/math] coefficients.
• If [math]k\leq 2[/math] then [math]\sum\lambda_i[/math] is a sum of cycles, with [math]\mathbb N[/math] coefficients.
• If [math]k\geq 3[/math] then [math]\sum\lambda_i[/math] might not decompose as a sum of cycles.
• [math]\sum\lambda_i[/math] has the same length as a sum of cycles: [math]N\in p_1\mathbb N+\ldots+p_k\mathbb N[/math].

Show Proof

This is something that we will not really need in what follows, but that we included here, in view of its importance. The idea of the proof is as follows:

(1) This is a well-known result, which follows from basic number theory, by using arguments in the spirit of those in the proof of Proposition 6.3.

(2) This is something that we already know at [math]k=1[/math], from Proposition 6.3. At [math]k=2[/math] the proof is more technical, along the same lines. See .

(3) The smallest possible [math]l[/math] potentially producing a counterexample is [math]l=2\cdot3\cdot 5=30[/math], and we have here indeed the sum given above, with [math]w=e^{2\pi i/30}[/math].

(4) This is a deep result, due to Lam and Leung, relying on advanced number theory knowledge. We refer to their paper for the proof.

■

As a side comment here, with such results we are now into rather advanced number theory. We warmly recommend at this point the reading of the paper of Lam-Leung , not that we will really need this in what follows, but for getting a taste of the subject. As a consequence now of the above result, we have the following generalization of the Butson obstruction, which is something final and optimal on this subject:

Theorem (Lam-Leung obstruction)

Assuming the we have

[[math]] l=p_1^{a_1}\ldots p_k^{a_k} [[/math]]

the following must hold, due to the orthogonality of the first [math]2[/math] rows:

[[math]] H_N(l)\neq\emptyset\implies N\in p_1\mathbb N+\ldots+p_k\mathbb N [[/math]]

In the case [math]k\geq2[/math], the latter condition is automatically satisfied at [math]N \gt \gt 0[/math].

Show Proof

Here the first assertion, which generalizes the [math]l=p^a[/math] obstruction from Proposition 6.4 above, comes from Theorem 6.9 (4), applied to the vanishing sum of [math]l[/math]-th roots of unity coming from the scalar product between the first 2 rows. As for the second assertion, this is something well-known, coming from basic number theory.

■

Summarizing, our study so far of the condition [math]H_N(l)\neq\emptyset[/math] has led us into an optimal obstruction coming from the first 2 rows, namely the Lam-Leung one, then an obstruction coming from the first 3 rows, namely the Sylvester one, and then two subtle obstructions coming from all [math]N[/math] rows, namely the de Launey one, and the Haagerup one. As an overall conclusion, by contemplating all these obstructions, nothing good in relation with our problem [math]H_N(l)\neq\emptyset[/math] is going on at small [math]N[/math]. So, as a natural and more modest objective, we should perhaps try instead to solve this problem at [math]N \gt \gt 0[/math].

The point indeed is that everything simplifies at [math]N \gt \gt 0[/math], with some of the above obstructions dissapearing, and with some other known obstructions, not to be discussed here, dissapearing as well. We are therefore led to the following statement:

\begin{conjecture}[Asymptotic Butson Conjecture (ABC)] The following equivalences should hold, in an asymptotic sense, at [math]N \gt \gt 0[/math],

• [math]H_N(2)\neq\emptyset\iff 4|N[/math],
• [math]H_N(p^a)\neq\emptyset\iff p|N[/math], for [math]p^a\geq3[/math] prime power,
• [math]H_N(l)\neq\emptyset\iff\emptyset[/math], for [math]l\in\mathbb N[/math] not a prime power,

modulo the de Launey obstruction, [math]|d|^2=N^N[/math] for some [math]d\in\mathbb Z[e^{2\pi i/l}][/math]. \end{conjecture} In short, our belief is that when imposing the condition [math]N \gt \gt 0[/math], only the Sylvester, Butson and de Launey obstructions survive. This is of course something quite nice, but in what regards a possible proof, this looks difficult. Indeed, our above conjecture generalizes the HC in the [math]N \gt \gt 0[/math] regime, which is so far something beyond reach. One idea, however, in dealing with such questions, coming from the de Launey-Levin result from , is that of looking at the partial Butson matrices, at [math]N \gt \gt 0[/math]. Observe in particular that restricting the attention to the rectangular case, and this not even in the [math]N \gt \gt 0[/math] regime, would make dissapear the de Launey obstruction from the ABC, which uses the orthogonality of all [math]N[/math] rows. We will discuss this later. For a number of related considerations, we refer as well to de Launey and de Launey-Gordon .

General references

Banica, Teo (2024). "Invitation to Hadamard matrices". arXiv:1910.06911 [math.CO].