Abstract Shalom and Tao showed that a polynomial upper bound on the size of single, large enough ball in Cayley graph implies underlying group has nilpotent subgroup with index degree growth both bounded effectively. The third fourth authors proved optimal this subgroup, at expense making some other parts result ineffective. In present paper, we prove without any losses elsewhere. As consequenc...

We analyse and compare the complexity of several algorithms for computing modular polynomials. We show that an algorithm relying on floating point evaluation of modular functions and on interpolation, which has received little attention in the literature, has a complexity that is essentially (up to logarithmic factors) linear in the size of the computed polynomials. In particular, it obtains th...

Consider a cubic extension K := Q(α), when the minimal polynomial f(x) of α does not totally split in K. The normal closure L := (K) is a sextic extension of Q, with Gal(L/Q) = S3. Now we fix notation and pick one embedding of K as K1, say K1 is fixed by (2, 3) ∈ S3. Here (2, 3) has the explicit description that if I choose one root u1 of f(x), (2, 3) permute the other two conjugate roots of u1...

The house of an algebraic integer of degree d is the largest modulus of its conjugates. For d ≤ 28, we compute the smallest house > 1 of degree d, say m(d). As a consequence we improve Matveev’s theorem on the lower bound of m(d). We show that, in this range, the conjecture of SchinzelZassenhaus is satisfied. The minimal polynomial of any algebraic integer α whose house is equal to m(d) is a fa...

Let f(x) 2 Z[x] be a monic irreducible reciprocal polynomial of degree 2d with roots r1, 1=r1, r2, 1=r2, . . . , rd, 1=rd. The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r1 +1=r1, . . . , rd +1=rd. If the Galois groups of f and g are Gf and Gg respectively, then Gg = Gf=N , where N is isomorphic to a subgroup of Cd 2 . In a naive sense, the generic case is...

The ring and polynomial learning with errors problems (Ring-LWE and Poly-LWE) have been proposed as hard problems to form the basis for cryptosystems, and various security reductions to hard lattice problems have been presented. So far these problems have been stated for general (number) rings but have only been closely examined for cyclotomic number rings. In this paper, we state and examine t...