2024/06/19 by Paolo Mantero, Vinh Nguyen, Mantero, Paolo +1 · 4 citations
Computer Science · #Advanced Algebra and Logic #Commutative Algebra (math.AC) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2406.13759
openalex publication_date 2024/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We describe the structure of the symbolic powers I(ℓ) of the Stanley-Reisner ideals, and cover ideals, I, of matroids. We (a) prove a structure theorem describing a minimal generating set for every I(ℓ); (b) describe the (non--standard graded) symbolic Rees algebra Rs(I) of I and show its minimal algebra generators have degree at most ht I; (c) provide an explicit, simple formula to compute the largest degree of a minimal algebra generator of Rs(I); (d) provide algebraic applications, including formulas for the symbolic defects of I, the initial degree of I(ℓ), and the Waldschmidt constant of I; (e) provide a new algorithm allowing fast computations of very large symbolic powers of I. One of the by-products is a new characterization of matroids in terms of minimal generators of I(ℓ) for some ℓ≥ 2. In particular, it yields a new, simple characterization of matroids in terms of the minimal generators of I(2). This is the first characterization of matroids in terms of I(2), and it complements a celebrated theorem by Minh-Trung, Varbaro, and Terai-Trung which requires the investigation of homological properties of I(ℓ) for some ℓ≥ 3.