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We show that the v -number of an arbitrary monomial ideal is bounded below by the v -number of its polarization and also find a criteria for the equality. By showing the additivity of associated primes of monomial ideals, we obtain the additivity of the v-numbers for arbitrary monomial ideals. We prove that the v -number v ( I ( G ) ) of the edge ideal I ( G ), the induced matching number im ( G ) and the regularity reg ( R / I ( G ) ) of a graph G , satisfy v ( I ( G ) ) ≤ im ( G ) ≤ reg ( R / I ( G ) ) , where G is either a bipartite graph, or a ( C 4 , C 5 ) -free vertex decomposable graph, or a whisker graph. There is an open problem in Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021), whether v ( I ) ≤ reg ( R / I ) + 1 , for any square-free monomial ideal I . We show that v ( I ( G ) ) > reg ( R / I ( G ) ) + 1 , for a disconnected graph G . We derive some inequalities of v -numbers which may be helpful to answer the above problem for the case of connected graphs. We connect v ( I ( G ) ) with an invariant of the line graph L ( G ) of G . For a simple connected graph G , we show that reg ( R / I ( G ) ) can be arbitrarily larger than v ( I ( G ) ) . Also, we try to see how the v -number is related to the Cohen–Macaulay property of square-free monomial ideals.

If the Krull dimension of the semigroup ring is greater than one, then affine semigroups of maximal projective dimension ( MPD ) are not Cohen–Macaulay, but they may be Buchsbaum. We give a necessary and sufficient condition for simplicial MPD -semigroups to be Buchsbaum in terms of pseudo-Frobenius elements. We give certain characterizations of ≺ -almost symmetric C -semigroups. When the cone is full, we prove the irreducible C -semigroups, and ≺ -almost symmetric C -semigroups with Betti-type three satisfy the extended Wilf conjecture. For e ≥ 4 , we give a class of MPD-semigroups in N 2 such that there is no upper bound on the Betti-type in terms of embedding dimension e . Thus, the Betti-type may not be a bounded function of the embedding dimension. We further explore the submonoids of N d , which satisfy the Arf property, and prove that Arf submonoids containing multiplicity are PI -monoids.

Let a and d be two linearly independent vectors in N 2 , over the field of rational numbers. For a positive integer k ≥ 2 , consider the sequence a , a + d , … , a + k d such that the affine semigroup S a , d , k = ⟨ a , a + d , … , a + k d ⟩ is minimally generated. We study the properties of affine semigroup ring K [ S a , d , k ] associated to this semigroup. We prove that K [ S a , d , k ] is always Cohen-Macaulay and it is Gorenstein if and only if k = 2 . For k = 2 , 3 , 4 , we explicitly compute the syzygies, the minimal graded free resolution and the Hilbert series of K [ S a , d , k ] . We also give a minimal generating set for the defining ideal of K [ S a , d , k ] which is also a Gröbner basis. Consequently, we prove that K [ S a , d , k ] is Koszul. Finally, we prove that the Castelnuovo–Mumford regularity of K [ S a , d , k ] is 1 for any a ,  d ,  k .

We show that the minimal number of generators and the Cohen-Macaulay type of a family of numerical semigroups generated by concatenation of arithmetic sequences is unbounded.

Conca and Varbaro (Invent. Math. 221 (2020), no. 3) showed the equality of depth of a graded ideal and its initial ideal in a polynomial ring when the initial ideal is square-free. In this paper, we give some beautiful applications of this fact in the study of Cohen-Macaulay binomial edge ideals. We prove that for the characterization of Cohen-Macaulay binomial edge ideals, it is enough to consider only \"biconnected graphs with some whisker attached\" and this done by investigating the initial ideals. We give several necessary conditions for a binomial edge ideal to be Cohen-Macaulay in terms of smaller graphs. Also, under a hypothesis, we give a sufficient condition for Cohen-Macaulayness of binomial edge ideals in terms of blocks of graphs. Moreover, we show that a graph with Cohen-Macaulay binomial edge ideal has girth less than \\(5\\) or equal to infinity.

We discuss some research problems on affine monomial curves, from the perspective of computation.

Let \\(\\) be the finite field with \\(q\\) elements. We study the number of \\(\\)-rational points on Danielewski and double Danielewski surfaces. For Danielewski surfaces, the point count is reduced to the number of roots of \\(P(Z)\\) over \\( \\) For double Danielewski surfaces, one has to count the number of tuples \\(( )ın^2\\), such that \\(P(0,)=0\\), \\(Q(0, )=0\\) hold simultaneously. We compute these numbers using gcd methods, resultants, character sums, Gauss sums, and the König--Rados theorem. We obtain explicit formulas in several structured cases, derive general bounds, and give a Macaulay2 algorithm for verification and show an intresting connection between the number of \\(\\)-rational points of these surfaces and polygonal numbers.

Let \\(CCM\\) denote the class of closed graphs with Cohen-Macaulay binomial edge ideals and \\(PIG\\) denote the class of proper interval graphs. Then \\(CCM PIG\\). The \\(PIG\\)-completion problem is a classical problem in molecular biology as well as in graph theory and this problem is known to be NP-hard. In this paper, we study the \\(CCM\\)-completion problem. We give a method to construct all possible \\(CCM\\)-completion of a graph. We find the \\(CCM\\)-completion number and the set of all minimal \\(CCM\\)-completions for a large class of graphs. Moreover, for that class, we give a polynomial-time algorithm to compute the \\(CCM\\)-completion number and a minimum \\(CCM\\)-completion of a given graph. We investigate unmixed and Cohen-Macaulay properties of binomial edge ideals of induced subgraphs. Also, we discuss the accessible graphs completion and the Cohen-Macaulay property of binomial edge ideals of whisker graphs.