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The relevance of Freiman’s theorem for combinatorial commutative algebra. (English) Zbl 1412.13025

Let \(X\) be a finite subset of \(\mathbb{Z}^n\) with affine hull \(A(X)\) and suppose \(d=\dim A(X)\). Freiman proves that \[|2X|\geq (d+1)|X|-\binom{d+1}{2}.\] A monomial ideal \(I\) of the polynomial ring \(S = \mathbb{K}[x_1,\ldots,x_n]\) is quasi-equigenerated if the exponent vectors of all generators of \(I\) belong to a hyperplane of \(\mathbb{Z}^n\). Using Freiman’s formula and its generalizations [K. Böröczky jun. et al., Discrete Comput. Geom. 52, No. 4, 705–729 (2014; Zbl 1310.11015)], it is shown as the first main result of the paper under review that for any quasi-equigenerated monomial ideal \(I\) with analytic spread \(\ell(I)\) and for every integer \(k\geq 1\), \[\mu(I^k)\geq \binom{\ell(I)+k-2}{k-1}\mu(I)-(k-1)\binom{\ell(I)+k-2}{k},\] where for a graded ideal \(J\), one denotes by \(\mu(J)\) the minimal number of generators of \(J\). The above inequality in the special case of \(k=2\) says \[\mu(I^2)\geq \ell(I)\mu(I)-\binom{\ell(I)}{2}.\] The authors call a quasi-equigenerated monomial ideal \(I\) a Freiman ideal, if the equality occurs in the above inequality. In other words, a quasi-equigenerated monomial ideal is a Freiman ideal, if the set of its exponent vectors achieves Freiman’s lower bound for its doubling. Equivalent conditions for a quasi-equigenerated monomial ideal \(I\) to be a Freiman ideal, in terms of vanishing of certain entries of the \(h\)-vector of the fiber cone \(F(I)\), are provided. Moreover, it is shown that \(I\) is Freiman if and only if \(F(I)\) is Cohen-Macaulay and its defining ideal has a \(2\)-linear resolution. Finally, finite simple graphs whose edge ideals or matroidal ideals of its cycle matroids are Freiman ideals are classified.

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13A15 Ideals and multiplicative ideal theory in commutative rings
13E15 Commutative rings and modules of finite generation or presentation; number of generators
13H05 Regular local rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

Citations:

Zbl 1310.11015
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References:

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