Cyclotomic and simplicial matroids

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corollary_2: For n = p_1^{m_1}⋯p_r^{m_r} the number of subsets of the n-th roots of unity that are Q-bases of the cyclotomic field is at most the product of p_i^{∏_{j≠i}(p_j−1)}, raised to the power p_1^{m_1−1}⋯p_r^{m_r−1}, with equality exactly when n = 2^a p^b q^c.

corollary_9: For n = p_1^{m_1}p_2^{m_2} with distinct primes p_1, p_2, the polynomial counting Q-linearly independent subsets I of the n-th roots of unity by y^{φ(n)−|I|} is the (p_1^{m_1−1}p_2^{m_2−1})-th power of an explicit coefficient of a logarithmic exponential generating function.

theorem_1: Martin and Reiner's main theorem: for n = p_1^{m_1}⋯p_r^{m_r} with distinct primes p_i and positive m_i, the matroid of the n-th roots of unity over Q is dual to the direct sum of p_1^{m_1−1}⋯p_r^{m_r−1} copies of the simplicial matroid of the join of r discrete vertex sets of sizes p_1, …, p_r.

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Jeremy L. Martin, Victor Reiner, "Cyclotomic and simplicial matroids," Israel J. Math. 150 (2005), 229-240. https://doi.org/10.1007/BF02762381 Preprint: arXiv:math/0402206 (2004).

Edition read. The copy read for this card, whole, is the arXiv preprint PDF (9 pages, printed pp. 1--9), which carries the stamp "arXiv:math/0402206v1 [math.CO] 12 Feb 2004" and prints no notice; the arXiv abstract page's license link points to arXiv's assumed license of 1991--2003 (http://arxiv.org/licenses/assumed-1991-2003/, from https://arxiv.org/abs/math/0402206v1, read 2026-10-02), every other right reserved.

Research digest

The paper identifies the rational linear-dependence matroid of the $n$-th roots of unity with the dual of a direct sum of $n/(p_1\cdots p_r)$ copies of a natural simplicial matroid determined by the distinct primes $p_1,\ldots,p_r$ dividing $n$ (Theorem 1, p. 2). With results of Bolker and Adin this bounds the number of subsets of the $n$-th roots of unity that are $\mathbb Q$-bases of $\mathbb Q(\zeta_n)$, with equality exactly when $n=2^ap^bq^c$ (Corollary 2, p. 3). When $n$ has two distinct prime factors $p_1,p_2$ that simplicial matroid is the graphic matroid of the complete bipartite graph $K_{p_1,p_2}$, so the roots-of-unity matroid is cographic (Remark 6, p. 5); in that case the paper also derives a generating function for the independent root subsets (Corollary 9, p. 7).

Result pages: Theorem 1 (p. 2, with Remarks 5 and 6), Corollary 2 (p. 3) and Corollary 9 (p. 7). Labels and pages are those of the arXiv preprint. Read status: claims checked for these statements, read clause by clause on the page images of the preprint; their proofs were read for structure only.

This is a useful dictionary for E0774: cyclotomic dependencies can sometimes be studied as cycles, cuts, or higher-dimensional boundaries. The limitation is important: the matroid records arbitrary rational dependence, while dissociation forbids only coefficients in $\{-1,0,1\}$. A matroid coloring argument may therefore be stronger than necessary and cannot be transferred without checking the coefficients of its circuits.

Bears on. E0774: indirectly, as a matroid dictionary for rational linear dependencies among roots of unity (Theorem 1) and as a count of the $\mathbb Q$-linearly independent, hence dissociated, sets of $n$-th roots of unity (Corollary 2 for bases, Corollary 9 for all sizes when $n$ has two prime factors); the paper proves nothing about dissociated sets of integers.

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