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Subalgebras generated in degree two with minimal Hilbert function. (English) Zbl 1473.13017

This paper is concerned with determining generators for subalgebras of polynomial rings with minimal Hilbert functions.
Recall that given a graded polynomial ring \(R\) over a field \(\mathbb{K}\) and a linearly independent subset \(W\subseteq R_d\) of homogeneous polynomials of degree \(d\) in \(R\), the Hilbert function of the graded subalgebra \(\mathbb{K}[W]\) is defined as \(\mathrm{HF}\left(\mathbb{K}[W],i\right)=\dim_\mathbb{K}\left(\mathrm{span} W^i\right)\).
A fundamental problem in combinatorial commutative algebra is to determine explicitly the subset \(W\) so that \(\mathrm{HF}\left(\mathbb{K}[W],i\right)\) takes the smallest possible value among all Hilbert functions of subalgebras generated by subsets of the same degree \(d\) and the same cardinality \(u\). Since the Hilbert function becomes a polynomial only for large values of the variable \(i\), such a Hilbert function will be minimal only in the asymptotic sense. In this context, minimality of the Hilbert function is equivalent to minimality of the degree and the leading coefficient of the respective Hilbert polynomial. This in turn is equivalent to minimizing the multiplicity of the respective subalgebra \(\mathbb{K}[W]\), which has a nice combinatorial description in the cases studied in this paper.
In [Rend. Ist. Mat. Univ. Trieste 50, 139–147 (2018; Zbl 1440.13070)], M. Boij and A. Conca proved the generating set \(W\) of the subalgebra with minimal Hilbert function will necessarily be a strongly stable set of monomials, meaning that for every monomial \(m\in W\) and every \(i<j\) such that \(x_j|m\) we must have \(x_im/x_j \in W\). In the special case that \(d=2\), strongly stable sets of monomials \(W\) can be associated to Ferrers graphs and the multiplicity of the subalgebra \(\mathbb{K}[W]\) can be computed by counting the number of maximal North-East paths in the respective Ferrers graph (Theorem 2.1). The author makes essential use of this result to study the minimality problem in the case \(d=2\). She conjectures (Conjecture 3.4) that minimality is achieved if the subalgebra is generated by the \(u\) greatest monomials of degree \(2\) with respect either to the Lexicographic or the Reverse Lexicographic monomial order. She then proceeds to verify the conjecture for a large class of cases and explicitly determines in which case Lex should be used versus RevLex (Theorem 3.8).
In particular, for small generating sets (\(u=|W|\leq 3240\)) the conjecture is verified computationally using Mathematica and the details are given in Appendix A. For larger values of \(u\) there are two subcases, one in which \(W\) is given by the initial Lex segment and one in which it is given by the initial RevLex segment. In both cases, the proof relies on a clever inductive argument (Proposition 3.9 and Lemma 3.13); the Lex case is relatively simpler, while RevLex requires a technical lemma on sums of binomial coefficients (Lemma 3.12) which the author uses to count paths on Ferrers graphs. In her concluding remarks (Section 4) the author proposes generalizations to degrees \(d\geq 3\) and argues using counterexamples that one would need to consider monomial orders different than Lex and RevLex as well. Finally, Appendix A contains the computational results for the small values of \(u\) mentioned above and Appendix B rephrases the Conjecture 3.4 in terms of combinatorics using integer partitions.

MSC:

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13C70 Theory of modules and ideals in commutative rings described by combinatorial properties
05E40 Combinatorial aspects of commutative algebra
14M12 Determinantal varieties

Citations:

Zbl 1440.13070
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References:

[1] Boij, M. and Conca, A., On Fröberg-Macaulay conjectures for algebras, Rend. Istit. Mat. Univ. Trieste 50 (2018), 139-147. https://doi.org/10.13137/2464-8728/22433 · Zbl 1440.13070 · doi:10.13137/2464-8728/22433
[2] Conca, A., Symmetric ladders, Nagoya Math. J. 136 (1994), 35-56. https://doi.org/10.1017/S0027763000024958 · Zbl 0810.13010 · doi:10.1017/S0027763000024958
[3] Corso, A. and Nagel, U., Specializations of Ferrers ideals, J. Algebraic Combin. 28 (2008), no. 3, 425-437. https://doi.org/10.1007/s10801-007-0111-2 · Zbl 1179.13008 · doi:10.1007/s10801-007-0111-2
[4] Corso, A., Nagel, U., Petrović, S., and Yuen, C., Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas, Forum Math. 29 (2017), no. 4, 799-830. https://doi.org/10.1515/forum-2016-0007 · Zbl 1373.13010 · doi:10.1515/forum-2016-0007
[5] Fröberg, R., An inequality for Hilbert series of graded algebras, Math. Scand. 56 (1985), no. 2, 117-144. https://doi.org/10.7146/math.scand.a-12092 · Zbl 0582.13007 · doi:10.7146/math.scand.a-12092
[6] Fröberg, R. and Lundqvist, S., Questions and conjectures on extremal Hilbert series, Rev. Un. Mat. Argentina 59 (2018), no. 2, 415-429. https://doi.org/10.33044/revuma.v59n2a10 · Zbl 1425.13007 · doi:10.33044/revuma.v59n2a10
[7] Grayson, D. R. and Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, Available at https://faculty.math.illinois.edu/Macaulay2/.
[8] Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL, 2018.
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