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Monomials, binomials and Riemann-Roch. (English) Zbl 1272.13017
In the paper under review, the authors state a Riemann-Roch theorem in the context of combinatorial commutative algebra. The role of curves is played by a special type of Artinian monomial ideals, called Riemann-Roch ideals, and the role of divisors is played by Laurent monomials. For a Riemann-Roch ideal \(M\), the authors give a notion of genus, canonical monomial \(\mathbf{x}^{\mathbf{K}}\) and rank of a Laurent monomial \(\mathbf{x}^{\mathbf{b}}\) with respect to \(M\) such that the Riemann-Roch formula \[ \text{rank}(\mathbf{x}^{\mathbf{b}}) - \text{rank}(\mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{b}}) = \text{degree}(\mathbf{x}^{\mathbf{b}}) - \text{genus}(M) + 1 \] holds.
The authors apply this technique to finite graphs \(G\) (connected, undirected and without loops) by means of the associated toppling ideal \(I_G\). First, they give a nice description of a set of minimal generators of \(I_G\), which turn out to be a Gröbner basis with respect to a reverse lexicographic order, and of a minimal resolution of \(\mathbb{K}[\mathbf{x}]/I_G\). Second, they prove that the initial ideal \(M_G\) of the toppling ideal \(I_G\) is Riemann-Roch, determining the canonical monomial and the genus of \(M_G\). Finally, they weaken the assumptions, allowing loops in the graph, in order to give a complete new proof of the Riemann-Roch formula for graphs firstly due to M. Baker and S. Norine [Adv. Math. 215, No. 2, 766–788 (2007; Zbl 1124.05049)].

MSC:
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
05C31 Graph polynomials
05E40 Combinatorial aspects of commutative algebra
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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