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We introduce the notion of a perfect path for a monomial algebra. We classify indecomposable non-projective Gorenstein-projective modules over the given monomial algebra via perfect paths. We apply the classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.

The reduction number of monomial ideals in the polynomial K[x, y] is studied. We focus on ideals I for which J = (xͣ, yb ) is a reduction ideal. The computation of the reduction number amounts to solve linear inequalities. In some special cases the reduction number can be explicitly computed.

Let K be a field, A a standard graded K-algebra and M a finitely generated graded A-module. Inspired by our previous works, see [2] and [3], we study the invariant called Hilbert depth of hM, that is hdepth(hM)=max{ d:∑j≤k(-1)k-j(d-jk-j)hM(j)≥0  for all k≤d }, \\mathrm{hdepth} \\left( {{h_M}} \\right) = \\max \\left\\{ {d:\\sum\\limits_{j \\le k} {{{\\left( { - 1} \\right)}^{k - j}}\\left( {\\matrix{ {d - j} \\cr {k - j} \\cr } } \\right){h_M}\\left( j \\right) \\ge 0\\,\\, \\mathrm{for}\\, \\mathrm{all}\\, k \\le d}} \\right\\}, where hM (−) is the Hilbert function of M , and we prove basic results regard it. Using the theory of hypergeometric functions, we prove that hdepth(hS) = n, where S = K[x1, . . . , xn]. We show that hdepth(hS/J ) = n, if J = (f1, . . . , fd) ⊂ S is a complete intersection monomial ideal with deg(fi) ≥ 2 for all 1 ≤ i ≤ d. Also, we show that hdepth(hM̅) ≥ hdepth(hM) for any finitely generated graded S-module M, where M̅ = M ⊗S S[xn+1].

In this paper we develop a theory of monomial preorders, which differ from the classical notion of monomial orders in that they allow ties between monomials. Since for monomial preorders, the leading ideal is less degenerate than for monomial orders, our results can be used to study problems where monomial orders fail to give a solution. Some of our results are new even in the classical case of monomial orders and in the special case in which the leading ideal defines the tangent cone.

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I ⊆ [x 0, … , xn ] we show that for all positive integers m, t and r, where e is the big-height of I and . This captures two conjectures (r = 1 and r = e): one of Harbourne and Huneke, and one of Bocci et al. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.

Whenever a numerical method produces accurate results, it creates an interesting functional equation, and because regularities is not assumed, unexpected solutions can emerge. Thus, this paper is mainly devoted to finding solutions to a generalized functional equation constructed in this spirit; namely, we solve the generalized form of the functional equation considered in Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012), then considered in Nadhomi et al. (Aequationes Math 95:1095–1117, 2021) and continued in Okeke and Sablik (Results Math 77:125, https://doi.org/10.1007/s00025-022-01664-x , 2022), that is we find the polynomial functions satisfying the following functional equation, 0.1 ∑ i = 1 n γ i F ( a i x + b i y ) = ∑ j = 1 m ( α j x + β j y ) f ( c j x + d j y ) , for every x , y ∈ R , γ i , α j , β j ∈ R , and a i , b i , c j , d j ∈ Q , and its special forms. Thus we continue investigations presented in Nadhomi et al. (Aequationes Math 95:1095–1117, 2021) where we generalized the left hand side of Fechner–Gselmann equation and those from Okeke and Sablik (Results Math 77:125, https://doi.org/10.1007/s00025-022-01664-x , 2022) where the right hand side of the Fechner–Gselmann equation was studied. It turns out that under some assumptions on the parameters involved, the pair ( F ,  f ) solving Eq. ( 0.1 ) happens to be a pair of polynomial functions.

We study the possible dynamical degrees of automorphisms of the affine space $\\mathbb {A}^n$ . In dimension $n=3$ , we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalizes the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\\mathbb {A}^n$ for some n, and we give the best possible n for quadratic integers, which is either $3$ or $4$ .

We study a class of finite groups, called almost monomial groups, which generalize the class of monomial groups and is connected with the theory of Artin L-functions. Our method of research is based on finding similarities with the theory of monomial groups, whenever it is possible.

Let $p$ be a prime, $G$ a solvable group and $P$ a Sylow $p$ -subgroup of $G$ . We prove that $P$ is normal in $G$ if and only if $\\unicode[STIX]{x1D711}(1)_{p}^{2}$ divides $|G:\\ker (\\unicode[STIX]{x1D711})|_{p}$ for all monomial monolithic irreducible $p$ -Brauer characters $\\unicode[STIX]{x1D711}$ of  $G$ .

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of ‘principal’ coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parameterizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multi-gradings in cluster algebras with principal coefficients. The properties of these parameterizations, some proven and some conjectural, suggest links to duality conjectures of Fock and Goncharov. The coefficient dynamics leads to a natural generalization of Zamolodchikov's $Y$-systems. We establish a Laurent phenomenon for such $Y$-systems, previously known in finite type only, and sharpen the periodicity result from an earlier paper. For cluster algebras of finite type, we identify a canonical ‘universal’ choice of coefficients such that an arbitrary cluster algebra can be obtained from the universal one (of the same type) by an appropriate specialization of coefficients.