Commutative algebra of generalised Frobenius numbers

Manjunath, Madhusudan; Smith, Ben

Algebraic Combinatorics, Tome 2 (2019) no. 1, pp. 149-171.

We study commutative algebra arising from generalised Frobenius numbers. The $k$-th (generalised) Frobenius number of relatively prime natural numbers $(a_1,\cdots ,a_n)$ is the largest natural number that cannot be written as a non-negative integral combination of $(a_1,\cdots ,a_n)$ in $k$ distinct ways. Suppose that $L$ is the lattice of integer points of ${(a_1,\cdots ,a_n)}^{\perp }$. Taking our cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modules $M_L^{\left(k\right)}$ whose Castelnuovo–Mumford regularity captures the $k$-th Frobenius number of $(a_1,\cdots ,a_n)$. We study the sequence ${\left\{M_L^{\left(k\right)}\right\}}_{k=1}^{\infty }$ of generalised lattice modules providing an explicit characterisation of their minimal generators. We show that there are only finitely many isomorphism classes of generalised lattice modules. As a consequence of our commutative algebraic approach, we show that the sequence of generalised Frobenius numbers forms a finite difference progression i.e. a sequence whose set of successive differences is finite. We also construct an algorithm to compute the $k$-th Frobenius number.

Classification : 11D07, 13D02, 52C07, 06A07
Mots clés : Frobenius number, syzygy, lattice, poset, Castelnuovo–Mumford regularity

[1] Alfonsín, J. L. Ramírez The Diophantine Frobenius problem, Oxford Lecture Series in Mathematics and its Applications, 30, Oxford University Press, 2005, xvi+243 pages | DOI | MR | Zbl

[2] Aliev, Iskander; De Loera, Jesús A.; Louveaux, Quentin Parametric polyhedra with at least $k$ lattice points: their semigroup structure and the $k$-Frobenius problem, Recent trends in combinatorics (The IMA Volumes in Mathematics and its Applications), Volume 159, Springer, 2016, pp. 753-778 | DOI | MR | Zbl

[3] Aliev, Iskander; Fukshansky, Lenny; Henk, Martin Generalized Frobenius numbers: bounds and average behavior, Acta Arith., Volume 155 (2012) no. 1, pp. 53-62 | DOI | MR | Zbl

[4] Amini, Omid; Manjunath, Madhusudan Riemann-Roch for sub-lattices of the root lattice $A_n$, Electr. J. Comb., Volume 17 (2010) no. 1, R124, 50 pages | Zbl

[5] Bayer, Dave; Stillman, Mike Computation of Hilbert functions, J. Symb. Comput., Volume 14 (1992) no. 1, pp. 31-50 | DOI | MR | Zbl

[6] Bayer, Dave; Sturmfels, Bernd Cellular resolutions of monomial modules, J. Reine Angew. Math., Volume 502 (1998), pp. 123-140 | DOI | MR | Zbl

[7] Beck, Matthias; Diaz, Ricardo; Robins, Sinai The Frobenius problem, rational polytopes, and Fourier-Dedekind sums, J. Number Theory, Volume 96 (2002) no. 1, pp. 1-21 | MR | Zbl

[8] Beck, Matthias; Robins, Sinai A formula related to the Frobenius problem in two dimensions, Number theory (New York, 2003), Springer, 2004, pp. 17-23 | MR | Zbl

[9] Beck, Matthias; Robins, Sinai Computing the continuous discretely. Integer-point enumeration in polyhedra, Undergraduate Texts in Mathematics, Springer, 2015, xx+285 pages | DOI | MR | Zbl

[10] Dilworth, Robert P. A decomposition theorem for partially ordered sets, Ann. Math., Volume 51 (1950), pp. 161-166 | DOI | MR | Zbl

[11] Eisenbud, David The geometry of syzygies. A second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics, 229, Springer, 2005, xvi+243 pages | MR | Zbl

[12] Grayson, Daniel R.; Stillman, Michael E. Macaulay2, a software system for research in algebraic geometry, available at https://faculty.math.illinois.edu/Macaulay2/

[13] Kannan, Ravi Lattice translates of a polytope and the Frobenius problem, Combinatorica, Volume 12 (1992) no. 2, pp. 161-177 | DOI | MR | Zbl

[14] Lorenzini, Dino Two-variable zeta-functions on graphs and Riemann-Roch theorems, Int. Math. Res. Not. (2012) no. 22, pp. 5100-5131 | DOI | MR | Zbl

[15] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Graduate Texts in Mathematics, 227, Springer, 2005, xiv+417 pages | MR | Zbl

[16] Peeva, Irena; Sturmfels, Bernd Generic lattice ideals, J. Am. Math. Soc., Volume 11 (1998) no. 2, pp. 363-373 | DOI | MR | Zbl

[17] Sabzrou, Hossein; Rahmati, Farhad The Frobenius number and $a$-invariant, Rocky Mt. J. Math., Volume 36 (2006) no. 6, pp. 2021-2026 | DOI | MR | Zbl

[18] Scarf, Herbert E.; Shallcross, David F. The Frobenius problem and maximal lattice free bodies, Math. Oper. Res., Volume 18 (1993) no. 3, pp. 511-515 | DOI | MR | Zbl

[19] Sturmfels, Bernd Gröbner bases and convex polytopes, University Lecture Series, 8, American Mathematical Society, 1996, xii+162 pages | MR | Zbl