The Frohman Kania-Bartoszynska ideal is an invariant associated to a 3-manifold with boundary and a prime p ≥ 5. We give some estimates of this ideal. We also calculate this invariant for some 3-manifolds constructed by doing surgery on a knot in the complement of another knot. Let p = 2d+ 1 ≥ 5 be a prime and let O = { Z[ζp] if p ≡ −1 (mod 4) , Z[ζp, i] = Z[ζ4p] if p ≡ 1 (mod 4) . If M is an o...

Given distinct points p1, . . . , pr of a smooth variety V (over an algebraically closed field k) and positive integers mi, Z = m1p1 + · · · +mrpr denotes the subscheme defined locally at each point pi by I mi i , where Ii is the maximal ideal in the local ring OV,pi at pi of the structure sheaf. More briefly, we say Z is a fat point subscheme of V . In the case that V is P for some n, it is of...

The derivative of an ideal in a number ring is defined and the relation between the ideal derivative and the arithmetic derivative of a number in Z is discussed. Some simple ideal di↵erential equations are also studied. Further, the definition of the ideal derivative is extended to the derivative of a fractional ideal in a number ring. Again, the relation between the fractional ideal derivative...

The GMRES and Arnoldi algorithms, which reduce to the CR and Lanczos algorithms in the symmetric case, both minimize p(A)b over polynomials p of degree n. The difference is that p is normalized at z 0 for GMRES and at z x for Arnoldi. Analogous "ideal GMRES" and "ideal Arnoldi" problems are obtained if one removes b from the discussion and minimizes p(/l)II instead. Investigation of these true ...

Let S = k[x1, . . . , xn] be a polynomial ring over a field k with n variables x1, . . . , xn, m the irrelevant maximal ideal of S, I a monomial ideal in S and I ′ the polarization of I in the polynomial ring S′ with ρ variables. We show that each graded piece H i m (S/I)a, a ∈ Z , of the local cohomology module H i m (S/I) is isomorphic to a specific graded pieceH i+ρ−n m ′ (S′/I )α, α ∈ Z , o...

We use Gröbner bases and a theorem of Handelman to show that an ideal I of R[x1, . . . , xk] contains a polynomial with positive coefficients if and only if no initial ideal inv(I), v ∈ R, has a positive zero. Let R = R[x1, . . . , xk], R = R[x1, . . . , xk] and, considering Laurent polynomials, let R̃ = R[x1 , . . . , xk ], R̃ = R[x ± 1 , . . . , x ± k ]. For a = (a1, . . . , ak) ∈ Z, write x = ...

Let M be a 2 and 3-torsion free prime Γ-ring, d a nonzero derivation on M and U a nonzero Lie ideal of M . In this paper it is proved that U is a central Lie ideal of M if d satisfies one of the following (i) d(U) ⊂ Z, (ii) d(U) ⊂ U and d(U) = 0, (iii) d(U) ⊂ U , d(U) ⊂ Z.

A as a subset of Q[X], so, if I is an ideal of A, then QI is the ideal of Q[X] generated by I. Given a ring R, we denote the natural mapping A → A ⊗ R = R[X] by ι R. This note is devoted to the following algorithmic problem (see [1] for a definition and properties of Gröbner bases of ideals in polynomial rings over Z). Problem (P). Suppose that we have an infinite sequence f 1 , f 2 ,. .. of el...

Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let SE denote the primes of E lying above those in S, and let OS E denote the ring of SE -integers of E. We then compare the Fitting ideal of K2(O E ) as a Z[G]-module with a higher Stickelberge...