OFFSET
1,6
COMMENTS
Poonen's abstract: The moduli space of rank-n commutative algebras equipped with an ordered basis is an affine scheme B_n of finite type over Z, with geometrically connected fibers. It is smooth if and only if n <= 3. It is reducible if n >= 8 (and the converse holds, at least if we remove the fibers above 2 and 3).
The relative dimension of B_n is (2/27) n^3 + O(n^{8/3}). The subscheme parameterizing etale algebras is isomorphic to GL_n/S_n, which is of dimension only n^2. For n >= 8, there exist algebras that are not limits of etale algebras. The dimension calculations lead also to asymptotic formulas for the number of commutative rings of order p^n and the dimension of the Hilbert scheme of n points in d-space for d >= n/2.
LINKS
Bjorn Poonen, The moduli space of commutative algebras of finite rank, version 2, Mar 21, 2007.
FORMULA
Array, by antidiagonals, A[k,n] = floor(prime(k)^((2/27)*(n^3))), where prime(k) = A000040(k).
EXAMPLE
Array begins:
=========================================================================
k.|.p_k.|.n=1.|.n=2.|.n=3.|.....n=4.|......n=5.|..........n=6.|.....n=7.|
=========================================================================
1.|.2...|...1.|...1.|...4.|......26.|......612.|........65536.|.4403251.|
2.|.3...|...1.|...1.|...9.|.....182.|....26169.|......4304672.|.........|
3.|.5...|...1.|...2.|..25.|....2058.|..2964452.|.152587890625.|.........|
4.|.7...|...1.|...3.|..49.|...10148.|.66831599.|..............|.........|
5.|.11..|...1.|...4.|.121.|...86491.|..........|..............|.........|
6.|.13..|...1.|...4.|.169.|..190948.|..........|..............|.........|
7.|.17..|...1.|...5.|.289.|..681144.|..........|..............|.........|
8.|.19..|...1.|...5.|.361.|.1154088.|..........|..............|.........|
9.|.23..|...1.|...6.|.529.|.2854950.|..........|..............|.........|
10|.29..|...1.|...7.|.841.|.8567414.|..........|..............|.........|
MAPLE
Digits := 100: for d from 1 to 10 do for n from 1 to d do k := d-n+1 ; A := floor(ithprime(k)^(2*n^3/27)) ; printf("%d, ", A) ; od: od: # R. J. Mathar, Jan 22 2009
CROSSREFS
KEYWORD
AUTHOR
Jonathan Vos Post, Nov 30 2008
EXTENSIONS
Corrected typo in A[4,3], reduced A[4,5] by 1, extended R. J. Mathar, Jan 22 2009
STATUS
approved