Mathematics

We introduce a novel construction which allows us to identify the elements of the skeletons of a CW-complex P(m) and the monomials in m variables. From this, we infer that there is a bijection between finite CW-subcomplexes of P(m) , which are quotients of finite simplicial complexes, and some bigraded standard Artinian Gorenstein algebras, generalizing previous constructions in \cite{F:S}, \cite{CGIM} and \cite{G:Z}. We apply this to a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicitly as a bigraded polynomial of bidegree (1,d) . We consider the algebra associated to polynomials of the same type of bidegree ( d 1 , d 2 ) .

The aim of this survey is to discuss invariants of Cohen-Macaulay local rings that admit a canonical module. Attached to each such ring R with a canonical ideal C, there are integers--the type of R, the reduction number of C--that provide valuable metrics to express the deviation of R from being a Gorenstein ring. We enlarge this list with other integers--the roots of R and several canonical degrees. The latter are multiplicity based functions of the Rees algebra of C. We give a uniform presentation of three degrees arising from common roots. Finally we experiment with ways to extend one of these degrees to rings where C is not necessarily an ideal.

Sets of zero-dimensional ideals in the polynomial ring k[x,y] that share the same leading term ideal with respect to a given term ordering are known to be affine spaces called Gröbner cells. Conca-Valla and Constantinescu parametrize such Gröbner cells in terms of certain canonical Hilbert-Burch matrices for the lexicographical and degree-lexicographical term orderings, respectively. In this paper, we give a parametrization of (x,y) -primary ideals in Gröbner cells which is compatible with the local structure of such ideals. More precisely, we extend previous results to the local setting by defining a notion of canonical Hilbert-Burch matrices of zero-dimensional ideals in the power series ring k[[x,y]] with a given leading term ideal with respect to a local term ordering.

We generalize Buchsbaum and Eisenbud's resolutions for the powers of the maximal ideal of a polynomial ring to resolve powers of the homogeneous maximal ideal over graded Koszul algebras. Our approach has the advantage of producing resolutions that are both more explicit and minimal compared to those previously discovered by Green and Mart\'ınez-Villa \cite{GreenMartinezVilla} or Mart\'ınez-Villa and Zacharia \cite{MartinezVillaZacharia}.

For a numerical semigroup ring K[H] we study the trace of its canonical ideal. The colength of this ideal is called the residue of H . This invariant measures how far is H from being symmetric, i.e. K[H] from being a Gorenstein ring. We remark that the canonical trace ideal contains the conductor ideal, and we study bounds for the residue. For 3 -generated numerical semigroups we give explicit formulas for the canonical trace ideal and the residue of H . Thus, in this setting we can classify those whose residue is at most one (the nearly-Gorenstein ones), and we show the eventual periodic behaviour of the residue in a shifted family.

Our main result is a construction of four families C_1,C_2,B_1,B_2 which are equipollent with the power set of the real line R and satisfy the following properties. (i) The members of the families are proper subfields of R whose algebraic closures equal the field C. (ii) Each field in C_1vC_2 contains a Cantor set. (iii) Each field in B_1vB_2 is a Bernstein set. (iv) All fields in C_1vB_1 are isomorphic. (v) If K,L are fields in C_2vB_2 then K is isomorphic to a subfield of L only in the trivial case K=L.

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound μ( I 2 )≥9 for the number of minimal generators of I 2 with μ(I)≥6 . Recently, Gasanova constructed monomial ideals such that μ(I)>μ( I n ) for any positive integer n . In reference to them, we construct a certain class of monomial ideals such that μ(I)>μ( I 2 )>⋯>μ( I n )=(n+1 ) 2 for any positive integer n , which provides one of the most unexpected behaviors of the function μ( I k ) . The monomial ideals also give a peculiar example such that the Cohen-Macaulay type (or the index of irreducibility) of R/ I n descends.

We consider the question of certifying that a polynomial in Z[x] or Q[x] is irreducible. Knowing that a polynomial is irreducible lets us recognise that a quotient ring is actually a field extension (equiv.~that a polynomial ideal is maximal). Checking that a polynomial is irreducible by factorizing it is unsatisfactory because it requires trusting a relatively large and complicated program (whose correctness cannot easily be verified). We present a practical method for generating certificates of irreducibility which can be verified by relatively simple computations; we assume that primes and irreducibles in F p [x] are self-certifying.

When do syzygies depend on the characteristic of the field? Even for well-studied families of examples, very little is known. For a family of random monomial ideals, namely the Stanley--Reisner ideals of random flag complexes, we prove that the Betti numbers asymptotically almost always depend on the characteristic. Using this result, we also develop a heuristic for characteristic dependence of asymptotic syzygies of algebraic varieties.

The notion of quasi f -ideals was first presented in [14] which generalize the idea of f -ideals. In this paper, we give the complete characterization of quasi f -ideals of degree greater or equal to 2 . Additionally, we show that the property of being quasi f -ideals remains the same after taking the Newton complementary dual of a squarefree monomial ideal I provided that the minimal generating set of I is perfect.