In this paper, binomial difference ideals are studied. Three canonical representations for Laurent binomial difference ideals are given in terms of the reduced Gröbner basis of Z[x]-lattices, regular and coherent difference ascending chains, and partial characters over Z[x]-lattices, respectively. Criteria for a Laurent binomial difference ideal to be reflexive, prime, well-mixed, and perfect a...

We define the monomial invariants of a projective variety Z; they are invariants coming from the generic initial ideal of Z. Using this notion, we generalize a result of Cook [C]: If Z is an integral variety of codimension two, satisfying the additional hypothesis sZ = sΓ, then its monomial invariants are connected.

Examples • If S = ∅, then Z(S) = An. • If S = {1}, then Z(S) = ∅. • If S = {x1 − a1, . . . , xn − an}, then Z(S) = {(a1, . . . , an)}. Remark. All rings in this course are commutative and have 1. Remark. If A is a ring, then any subset S ⊆ A generates a minimal ideal 〈S〉 ⊆ A. In fact, we have 〈S〉 = {∑ aj xj : aj ∈ A, xj ∈ S}. Lemma. Z(S) = Z(〈S〉) for all S ⊆ k[x1 , . . . , xn]. Proof. Since 〈S〉...

Let p > 3 be a prime, ζp a pth primitive root of 1, and ∆ the Galois group of Q(ζp) over Q. Let q 6= p be a prime and n the order of q modulo p. Assume q 6≡ 1 mod p and so n ≥ 2, p(q− 1)|qn − 1, and n|p− 1. Set f = (q − 1)/p and e = (p− 1)/n. Let Q be a prime ideal of Z[ζp] above q and let F = Z[ζp]/Q. Thus F ∼= Fqn , the finite field with q elements. Let α ∈ Z[ζp] be a generator of F such that...

If E is a subset of Z then the n-th characteristic ideal of the algebra of rational polynomials taking integer values on E, Int(E, Z), is the fractional ideal consisting of 0 and the leading coefficients of elements of Int(E, Z) of degree no more than n. For p a prime the characteristic sequence of Int(E, Z) is the sequence of negatives of the p-adic values of these ideals. We give recursive fo...

Let Z be a fat point scheme in P supported on general points. Here we prove that if the multiplicities are at most 3 and the length of Z is sufficiently high then the number of generators of the homogeneous ideal IZ in each degree is as small as numerically possible. Since it is known that Z has maximal Hilbert function, this implies that Z has the expected minimal free resolution.

A bijection is proved between Sl( n, Z)-conjugacy classes of hyperbolic matrices with eigenvalues {A,,_A,,} which are units in an n-degree number field, and narrow ideal classes of the ring Rk = Z[A,]. A bijection between Gl(«,Z)-conjugacy classes and the wide ideal classes, which had been known, is repeated with a different proof. In 1980, Peter Sarnak was able to obtain an estimate of the gro...

In this talk, we introduce Hilbert functions of a graded algebras over a field, and one of the long standing conjectures concerning them, Fröberg conjecture. Then we study relations between Fröberg conjecture on Hilbert series and Moreno-Socias Conjecture. Consequently, we show that Fröberg conjecture holds for special cases as an example. 1. Almost Reverse Lexicographic Monomial Ideal Let R = ...