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Multi-Rees algebras and toric dynamical systems. (English) Zbl 1429.13006

In this well-written article, the authors establish a connection between Rees algebras in commutative algebra and chemical reaction networks.
Given a commutative ring \(R\) and a collection of ideals \(I_1,\ldots, I_l\subset R\), the multi-Rees algebra of \(I_1,\ldots, I_l\) is the multigraded \(R\)-algebra \[ \mathcal R_R(I_1\oplus \cdots \oplus I_l) = \bigoplus_{a_1,\ldots, a_l\geq 0} I_1^{a_1}\cdots I_l^{a_l}t_1^{a_1}\cdots t_l^{a_l}\subset R[t_1,\ldots, t_l], \] where \(R[t_1,\ldots, t_l]\) is the polynomial ring with coefficients in \(R\). For \(l=1\), the multi-Rees algebra is the usual Rees algebra of an ideal.
A chemical reaction network is modeled by a directed graph \(G\): each directed edge in \(G\) represents a chemical or biochemical reaction. Each vertex of \(G\) supports an exponent vector that encodes how the vertex is produced from the molecules or species involved in the reaction. From \(G\), one constructs a toric ideal \(T_G\).
A directed graph \(G\) with vertex set \(V\) is strongly connected if, given any two vertices \(i,j\in V\), there is a directed path from \(i\) to \(j\) and a directed path from \(j\) to \(i\). With each strongly connected component of \(G\), one associates a monomial ideal built from the vertices in that component.
The main result (Theorem 3.3) is as follows. Let \(G\) be a directed graph, with \(l\) strongly connected components, associated to a chemical reaction. Then its toric ideal \(T_G\) is the ideal defining the multi-Rees algebra \(\mathcal R_R(I_1\oplus \cdots \oplus I_l)\), where \(I_j\) is the monomial ideal corresponding to the \(j\)-th strongly connected component of \(G\).

MSC:

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13F65 Commutative rings defined by binomial ideals, toric rings, etc.
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
92C42 Systems biology, networks
37N25 Dynamical systems in biology
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References:

[1] Botbol, Nicol\'{a}s, The implicitization problem for \(\phi\colon\mathbb{P}^n \dashrightarrow(\mathbb{P}^1)^{n+1} \), J. Algebra, 322, 11, 3878-3895 (2009) · Zbl 1185.14008 · doi:10.1016/j.jalgebra.2009.03.006
[2] Bruns, Winfried; Conca, Aldo, Linear resolutions of powers and products. Singularities and computer algebra, 47-69 (2017), Springer, Cham · Zbl 1405.13022
[3] Craciun, Gheorghe; Dickenstein, Alicia; Shiu, Anne; Sturmfels, Bernd, Toric dynamical systems, J. Symbolic Comput., 44, 11, 1551-1565 (2009) · Zbl 1188.37082 · doi:10.1016/j.jsc.2008.08.006
[4] Dickenstein, Alicia, Biochemical reaction networks: an invitation for algebraic geometers. Mathematical Congress of the Americas, Contemp. Math. 656, 65-83 (2016), Amer. Math. Soc., Providence, RI · Zbl 1346.13063 · doi:10.1090/conm/656/13076
[5] Dickenstein, Alicia; Di Rocco, Sandra; Piene, Ragni, Classifying smooth lattice polytopes via toric fibrations, Adv. Math., 222, 1, 240-254 (2009) · Zbl 1193.14065 · doi:10.1016/j.aim.2009.04.002
[6] edelstein B. Edelstein, Biochemical models with multiple steady states and hysteresis, J. Theor. Biol. 29 (1970), 57-62.
[7] feinberglectures M. Feinberg, Lectures on chemical reaction networks, 1979, https://crnt.osu.edu/LecturesOnReactionNetworks.
[8] Gatermann, Karin; Wolfrum, Matthias, Bernstein’s second theorem and Viro’s method for sparse polynomial systems in chemistry, Adv. in Appl. Math., 34, 2, 252-294 (2005) · Zbl 1075.65074 · doi:10.1016/j.aam.2004.04.003
[9] Ito, Atsushi, Algebro-geometric characterization of Cayley polytopes, Adv. Math., 270, 598-608 (2015) · Zbl 1333.14048 · doi:10.1016/j.aim.2014.11.010
[10] Jabarnejad, Babak, Equations defining the multi-Rees algebras of powers of an ideal, J. Pure Appl. Algebra, 222, 7, 1906-1910 (2018) · Zbl 1393.13007 · doi:10.1016/j.jpaa.2017.08.013
[11] Lin, Kuei-Nuan, Cohen-Macaulayness of Rees algebras of modules, Comm. Algebra, 44, 9, 3673-3682 (2016) · Zbl 1360.13014 · doi:10.1080/00927872.2015.1086925
[12] SosaMulti G. Sosa, On the Koszulness of multi-Rees algebras of certain strongly stable ideals, arXiv:1406.2188[math.AC], 2014.
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