Roots-of-unity decomposition conjecture for generalized Vandermonde matrices

About 22 years old · traced to 2004

Let $X=\{x_0=1,x_1,\ldots,x_k\}$ be a set of pairwise distinct roots of unity, and let $B=\{b_0=0,b_1,\ldots,b_m\}$ be a set of coprime integer numbers. Let $M=(x_i^{b_j})$ be degenerate. Roots-of-unity decomposition conjecture. Then $B$ can be split as

B=B_1\sqcup\cdots\sqcup B_k,

such that all the $x_i$ are roots of unity of degree

\gcd\{b-b'\mid b,b'\in B_i,\ i=1,\ldots,k\}.

The statement concerns the structure of degeneracy loci of generalized Vandermonde matrices. The paper says that it proves this conjecture for $k\leqslant2$ , while the arbitrary- $k$ formulation remains open.

References

Primary source

Alexander Esterov, Evgeny Statnik and Arina Voorhaar, “Sparse curve singularities, singular loci of resultants, and Vandermonde matrices”, arXiv:2301.07001 (2023).

Additional references

2 papers in this index state this conjecture (2004–2023). The statement above is taken from the most recent of them; the others are arXiv:math/0407306.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Open

The conjecture is proved for configurations involving at most three roots of unity, but the general higher-dimensional case remains open.

The conjecture predicts a precise decomposition of the exponent set whenever a generalized Vandermonde matrix built from distinct roots of unity is degenerate. The paper Sparse curve singularities, singular loci of resultants, and Vandermonde matrices, published on January 17, 2023, formulates the arbitrary- $k$ problem and leaves that case open.

Known results

The conjecture is proved for $k\leqslant 2$ , including the three-root case characterized in Lemma 5.22 (2023).

Current status (as of August 2026): The cases $k\leqslant 2$ are settled, while the arbitrary- $k$ conjecture remains open with no reported proof, counterexample, or verification.

Sources

No solutions have been posted yet.