LINEAR RELATIONS AMONG GALOIS CONJUGATES OVER $\mathbb{F}_{q}(t)$

We classify the coefficients $(a_{1},...,a_{n})\in\mathbb{F}_{q}[t]^{n}$ that can appear in a linear relation $\sum_{i=1}^{n}a_{i}\gamma_{i}=0$ among Galois conjugates $\gamma_{i}\in\overline{\mathbb{F}_{q}(t)}$. We call such an $n$-tuple a Smyth tuple. Our main theorem gives an affirmative answer to a function field analogue of a 1986 conjecture of Smyth [4] over $\mathbb{Q}$. Smyth showed that certain local conditions on the $a_{i}$ are necessary and conjectured that they are sufficient. Our main result is that the analogous conditions are necessary and sufficient over $\mathbb{F}_{q}(t)$, which we show using a combinatorial characterization of Smyth tuples from [4]. We also formulate a generalization of Smyth’s Conjecture in an arbitrary number field that is not a straightforward generalization of the conjecture over $\mathbb{Q}$ due to a subtlety occurring at the archimedean places.

The question of how prevalent linear relations among Galois conjugates are has been studied from multiple angles. In [2] it is shown that for a Hilbertian field $K$ and all but finitely many nonnegative integers $n$, there exists an algebraic number $\alpha\in\overline{K}$ of degree $2^{n}n!$ whose conjugates span a vector space of dimension $n$. In these cases, the dimension of relations between conjugates is $2^{n}n!-n$, and so in this sense, linear relations among conjugates are plentiful.

In an older paper, Smyth showed [4, Cor. 2] that for any $a_{1},\dots,a_{n}\in{\mathbb{Z}}$ with $gcd(a_{1},\dots,a_{n})=1$, if there exist Galois conjugates $\gamma_{1},\dots,\gamma_{n}\in\overline{\mathbb{Q}}$ such that $\sum_{i=1}^{n}a_{i}\gamma_{i}=0$, then the $a_{i}$ necessarily satisfy the following two properties.

1. (1)

   $|a_{i}|\leq\sum_{j\neq i}|a_{j}|$ for all $i$

2. (2)

   Every prime $p$ divides at most $n-2$ of the $a_{i}$.

Smyth conjectured that the converse is true as well over $\mathbb{Q}$.

A natural way to generalize (1) and (2) to an arbitrary field $K$ is as follows.

• (1’)

  For any archimedean absolute value $|\cdot|$ of $K$, we have $|a_{i}|\leq\sum_{j\neq i}|a_{j}|$ for all $i$.

• (2’)

  For any nonarchimedean absolute value $|\cdot|$ of $K$, we have $|a_{i}|\leq\max_{j\neq i}|a_{j}|$ for all $i$.

We call (1’) and (2’) the absolute value criteria over $K$.

Note that for any $a\in K^{*}$, a tuple $(a_{1},...,a_{n})\in K^{n}$ satisfies the absolute value criteria if and only if $(aa_{1},...,aa_{n})$ does, and similarly, $(a_{1},...,a_{n})$ is a Smyth tuple if and only if $(aa_{1},...,aa_{n})$ is a Smyth tuple. Hence, when $K$ is the field of fractions of a principal ideal domain, it is enough to look at coprime tuples $(a_{1},...,a_{n})$.

Conjecture 2.1 remains open. Our main result is Theorem 4.2, which answers in the affirmative the natural analogue of Smyth’s question over ${\mathbb{F}}_{q}(t)$. We will also formulate in Conjecture 10.1 a generalization of Conjecture 2.1 for arbitrary number fields, which involves a subtlety not present in the cases of $\mathbb{Q}$ or $\mathbb{F}_{q}(t)$. A MathOverflow post of David Speyer [1] proves a special case of this conjecture.

In Section 3 we record some terminology that we will use throughout the paper. In Section 4 we set up Theorem 4.2, which we go on to prove in Section 5. Section 6 examines how close the Galois relations constructed in our proof of Theorem 4.2 are to “as small as possible.” In Section 7 we show that the conjugates in our constructed linear relations for Smyth triples can be chosen to have the full symmetric group as their Galois group and in Section 8 we give heuristic reasoning for why we should expect the Galois group of linear Galois relations to be large in general. In Section 9, we record some results from Section 6 that also apply over $\mathbb{Q}$. Finally, in Section 10, we formulate a number field generalization of Conjecture 2.1, and then reduce it slightly to Conjecture 10.7.

We lay out some terminology and a background result.

We say that $\gamma_{1},...,\gamma_{n}$ are Galois conjugates over K if they are all roots of the same irreducible polynomial over $K$. (When the base field $K$ is clear from the context, we will often omit it from our terminology.) Throughout the paper, we assume Galois conjugates are nonzero.

A linear Galois relation will mean a linear relation $\sum_{i=1}^{n}a_{i}\gamma_{i}=0$ among Galois conjugates $\gamma_{1},...,\gamma_{n}\in\overline{K}$ with coefficients $a_{i}\in K$.

We will call a tuple $(a_{1},...,a_{n})\in\mathbb{F}_{q}[t]^{n}$ coprime if $a_{1},...,a_{n}$ generate the unit ideal (and similarly for $(a_{1},...,a_{n})\in\mathbb{Z}^{n}$). The (logarithmic) height of a coprime tuple $(a_{1},...,a_{n})\in\mathbb{F}_{q}[t]^{n}$ is $\max_{i}\deg(a_{i})$. The (logarithmic) height of a coprime tuple $(a_{1},...,a_{n})\in\mathbb{Z}^{n}$ is $\max_{i}\log|a_{i}|$. A coprime tuple is said to be a Smyth tuple if its coordinates appear as the coefficients of a linear Galois relation.

A balanced multiset of tuples – which we will sometimes shorten to “balanced multiset” – with respect to a tuple $(a_{1},...,a_{n})$ is a nonempty collection of nonzero solutions $\{(x_{i_{1}},...,x_{i_{n}})\in K^{n}\}_{i=1}^{N}$ to the equation $\sum_{i=1}^{n}a_{i}x_{i}=0$ such that the multiset $\{x_{i_{j}}\}_{i=1}^{N}$ is independent of $j$; in other words, for all $x\in K$, $x$ appears in each of the $N$ coordinate positions the same number of times.

A balanced multiset $\{(x_{i_{1}},...,x_{i_{n}})\in K^{n}\}_{i=1}^{N}$ is a 1-factor if the multiset $\{x_{i_{1}}\}_{i=1}^{N}$ is in fact a set. (To see the justification for this terminology, consider the hypergraph whose vertices are elements of $K$ and whose edges are ordered tuples $(x_{1},...,x_{n})$ of solutions to $\sum_{i=1}^{n}a_{i}x_{i}=0$. Then a 1-factor in this hypergraph is precisely our definition of 1-factor.)

The degree of a linear Galois relation $\sum_{i=1}^{n}a_{i}\gamma_{i}=0$ will refer to the degree $[K(\gamma_{1}):K]$ of the conjugates in the linear relation. Similarly the Galois group of a linear Galois relation $\sum_{i=1}^{n}a_{i}\gamma_{i}=0$ will mean $\operatorname{Gal}(E/K)$, where $E$ is the Galois closure of $K(\gamma_{1})$.

Following standard terminology, we will call a Galois extension $L/K$ a $G$-extension if $\operatorname{Gal}(L/K)\cong G$.

Our final preliminary is the fact that over any Hilbertian field, there exist $S_{d}$-extensions for all $d\geq 1$. This is a special case of [2, Prop. 8]. We will apply this fact to the Hilbertian field $\mathbb{F}_{q}(t)$.

Although our primary focus in this section (and this paper) is the function field $\mathbb{F}_{q}(t)$, we will often state definitions and theorems over an arbitrarily field $K$ when it is not substantially more complicated to do so.

Our approach in this paper benefits from having several equivalent notions of Smyth tuples. Most importantly, Smyth proved a combinatorial characterization of Smyth tuples involving balanced multisets. We record this, along with one other characterization of Smyth tuples, in Proposition 4.1.

Our main theorem is the following.

The forward direction of Theorem 4.2 was established by Smyth over $\mathbb{Q}$ in a proof that is valid over a general field $K$. We record this result in Lemma 4.3.

We now turn toward the proof of our main result, Theorem 4.2, which will follow essentially as a corollary from Proposition 5.1.

Let $V_{N}\subset{\mathbb{F}}_{q}[t]$ be the set of polynomials of degree $<N$. We will show that, surprisingly, for any coprime $(a_{1},..,a_{n})\in\mathbb{F}_{q}[t]^{n}$ satisfying the absolute value criteria and any $N\geq d=\max\deg_{i}a_{i}$, the set of all solutions $(x_{1},...,x_{n})\in V_{N}^{n}$ to the equation $\sum_{i=1}^{n}a_{i}x_{i}=0$ is a balanced set.

Now Theorem 4.2 follows easily.

In the previous section, we produced balanced multisets for all Smyth tuples, i.e. all tuples for which it is possible to find a balanced multiset. We relied on Proposition 4.1 to turn the balanced multisets into linear Galois relations. In this section, we consider the question of how close these linear Galois relations are to “as small as possible.” As a first attempt to formulate this question, we might ask for a uniform lower bound $\ell_{n}(D)$ on the degree of all linear Galois relations whose coefficients are an $n$-tuple of height $D$. However, this is quickly seen to be an uninteresting question, as there exist Smyth tuples of arbitrarily large height that are coefficients of a bounded degree linear Galois relation. (For instance, if $\sum_{i=1}^{n}\epsilon_{i}a_{i}=0$ where $\epsilon_{i}\in\{1,-1\}$, then there is a balanced multiset with respect to $(a_{1},...,a_{n})$ of size $2$.) So instead, we ask for a lower bound $L_{n}(D)$ on the “worst” Smyth $n$-tuple of height $D$.

In this section the underlying field $K$ will always be $\mathbb{F}_{q}(t)$, and so we will omit it from the notation. In Section 9 we will record analogous results about $L^{\mathbb{Q}}_{3}(D)$.

In Corollary 6.4, we prove that there exist Smyth triples $(a_{1},a_{2},a_{3})$ of height $D$ for which one cannot find a balanced multiset of size less than $|V_{D}\setminus\{0\}|=q^{D}-1$. In other words, besides removing zero from the multiset, our construction in Section 4 cannot be uniformly improved upon.

In Proposition 6.2, we (non-constructively) show that a $3$-term linear Galois relation of degree $d$ with coefficients $(a_{1},a_{2},a_{3})$ implies the existence of a balanced multiset with respect to $(a_{1},a_{2},a_{3})$ of size $d$. Thus we obtain in Corollary 6.5 the lower bound $q^{D}-1\leq L_{3}(D)$.

Note that this does not imply that the linear Galois relations arising from our balanced multisets in the $n=3$ case have degree as small as possible; the linear Galois relations arising from a balanced multiset of size $q^{D}-1$ may have degree as large as $(q^{D}-1)!$ (see Theorem 7.2).

In this section we show that the conjugates in our construction for Smyth triples from Section 5 can be chosen to have the full symmetric group as their Galois group.

We begin with a lemma.

We just saw in Section 7 that the conjugates appearing in the linear Galois relations we constructed for Smyth triples $(a,b,c)\in\mathbb{F}_{q}[t]^{3}$ can be chosen to generate an $S_{q^{d}}$-extension of $\mathbb{F}_{q}(t)$ where $d$ is the height of $(a,b,c)$. In this section, we consider which Galois groups we expect to arise in this context.

Heuristic 8.3, which depends only on $|G|$ and $m$, will correctly predict that Smyth tuples $(a_{1},...,a_{n})$ are always $S_{q^{N}}$-tuples for sufficiently large $N$. (Recall that Remark 5.2 and Theorem 7.2 together show that every Smyth triple is an $S_{q^{N}}$-triple for sufficiently large $N$.)

By Proposition 4.1, a coprime $n$-tuple $(a_{1},...,a_{n})\in\mathbb{F}_{q}[t]^{n}$ of height $d$ is a Smyth tuple if and only if there exist permutation matrices $X_{1},...,X_{n}$ such that $\det(\sum_{i=1}^{n}a_{i}X_{i})=0$. Our approach so far has been to look for balanced multisets, which is essentially asking, given $(a_{1},...,a_{n})$, whether or not there exists a vector $v\in\mathbb{F}_{q}[t]^{n}$ and permutation matrices $X_{1},..X_{n}$ such that $(\sum_{i=1}^{n}a_{i}X_{i})v=0$. We now shift our perspective slightly by fixing both the tuple $(a_{1},...,a_{n})$ and a vector $v$, and asking whether or not there exist such permutation matrices in $G$. Since $v$ will have no repeated entries, the heuristic is suited only to detect 1-factors, as opposed to a general balanced multiset.

In particular, we will fix a vector $v_{N}\in\mathbb{F}_{q}[t]^{q^{N}}$ whose coordinates are all the elements of $V_{N}$ in some order, each occurring exactly once. We have already seen that for any Smyth triple $(a_{1},a_{2},a_{3})$, there do exist permuation matrices $X_{1},X_{2},X_{3}\in S_{q^{N}}$ such that $(a_{1}X_{1}+a_{2}X_{2}+a_{3}X_{3})v_{N}=0$ (Remark 5.2). Our heuristic, in the case of triples, suggests an answer to the question of whether this remains true if we insist on choosing $X_{1},X_{2},X_{3}$ from some subgroup $G\subset S_{q^{N}}$.

The result of this assumption is that we have $\sum_{i=1}^{n}(a_{i}X_{i})v=0$ with probability $(q^{-(N+d)})^{q^{N}}$. On the other hand, we have $|G|^{n-1}$ choices for $X_{1},...,X_{n-1}$, so, according to this model, the probability of failing to find $X_{1},...,X_{n}$ such that $\sum_{i=1}^{n}(a_{i}X_{i})v=0$ is $p_{N}:=(1-q^{-(N+d)q^{N}})^{|G|^{(n-1)}}$.