Maybe Christmas is not a good time to float the indigestible screed below --- but it's what I've got on the brain at the moment, so here goes! WFL Companion manifolds --- I am interested in a certain family of algebraic manifolds, call them J(S), with components ranging over some finite field \F_q. The question arises, given that J(S) is a singular scroll of dimension 2, can the number of points on it be bounded in terms of q? What can be said more generally about its geometry? I have so far explored J(S) for 2 points S over \F_2, having 6,8 points more (besides S) resp; and 1 point over \F_3, having 24 points more. [OK, I know this doesn't look very impressive --- but the software was a pig to develop!] Definition of J(S) --- {X_ij | i,j in \Z} denotes a two-dimensional array of variables; S = [S_ij], T, U denote points in the corresponding \N-dimensional projective space I. A^ij denote "cruciform" matrices such that (A^ij)_kl = 1 for (k,l) = (i,j); -1 for (k,l) = (i-1,j), (i+1,j), (i,j-1), (i,j+1); 0 otherwise, for k,l in \Z. K denotes the \N-dimensional manifold intersecting the set of quadrics {T' (A^ij) T = 0 | i,j in \Z}; given some point S on K, J(S) denotes the manifold intersecting K with the set of flats {S' (A^ij) T = 0 | i,j in \Z} [where X' denotes transpose]. Lemma --- J(S) is the union of (all points on) a set of lines through its "vertex" S. For if X = aS + bT is an arbitrary point on the line joining S,T, easily X' (A^ij) X = a^2 S' (A^ij) S + 2ab(S' (A^ij) T) + b^2 T' (A^ij) T = 0, so X also lies on J(S). Syzygy --- Among the "2-diamond" set of 15 equations in 26 variables {S' (A^ij) S = 0 | (i,j) = (0,0), (-1,0), (0,-1), (+1,0), (0,+1)} U {T' (A^ij) T = 0 | (i,j) = (0,0), (-1,0), (0,-1), (+1,0), (0,+1)} U {S' (A^ij) T = 0 | (i,j) = (0,0), (-1,0), (0,-1), (+1,0), (0,+1)} any one is implied by the remaining 14. [The syzygy relation is omitted.] Geometrically, it follows for instance that a line lying on four such quadrics, and passing through 2 points of the fifth, must also lie on the fifth. Theorem --- J(S) has dimension two. Proof: Given n in \N, consider the finite "n-diamond" section of I defined by {X_ij | |i|+|j| < n}; the number of components in this set equals m(n) = (1 + 3 + ... + (2n+1) + ... + 3 + 1) = 2(n+1)n + 1; the number of quadrics (1-diamonds) involving only these components equals m(n-1); the number of 5-quadric sets (2-diamonds) equals m(n-2). Within this section, an arbitrary point has freedom d = m(n)-1, losing 1 for each quadric to which it is constrained; so a point on K has freedom e = m(n)-1 - m(n-1). An arbitrary line has freedom 2(d-1), losing 3 for each quadric to which it is constrained, but gaining 1 for each 2-diamond via the syzygy; so a line on K has freedom f = 2m(n)-4 - 3m(n-1) + m(n-2). Finally, the dimension of J(S) equals f+1 - e = 2m(n)-4 - 3m(n-1) + m(n-2) + 1 - m(n)+1 + m(n-1) = m(n) - 2m(n-1) + m(n-2) - 2 = 4-2 = 2. Now let n -> oo. QED. Critique --- A number of features of the argument above are of dubious provenance. In what sense can the space I and manifold K be said to exist at all, as entities on which it might be legitimate to perform algebraic geometry? The ground field is discrete; and even if it were continuous, there is no apparent metric (as is available for example in Hilbert space). In particular, under what circumstances is it possible to justify letting n -> oo ? In particular cases the denumerable dimension can be circumvented: for instance, if S is singly- or doubly-periodic, the array of variables may be wrapped around into a cylinder or torus; and if the points involved have zero components for sufficently large |i| or |j| or both, the array may be truncated at that bound. Plainly, such stratagems are theoretically undesirable. In any case, bearing in mind the translation symmetry of the equation system for K, it in practice usually proves sufficient to consider a single 2-diamond. Again, the argument implicitly assumes that all points S on K are equivalent. Each quadric is invariant under some conjugate of a mixed orthogonal group; but it's far from clear whether the intersection of these is transitive on the points of K. Motivation --- A "number wall" is most easily thought of as a difference table on steroids, allowing an LFSR sequence to be detected, interpolated etc. in much the same way as the latter serves a polynomial sequence: in this initial incarnation, the number wall is the 2-D array of persymmetric determinants formed from sub-blocks of the given sequence. The 2001 paper gave identities permitting recursive numerical computation of a number wall in the presence of square "windows" of zero components; for these "frame theorems" it provided an involved and delicate inductive proof relying on a deal of intrusive analytical machinery. A purely algebraic approach circumvents most of this complexity, as well as tying up a number of loose ends in the earlier treatment. The price to be paid is the transmogrification of a number wall (modulo a constant factor) into a point S in the alarmingly large manifold K above. Having grasped this nettle, we are rewarded in an astonishing fashion. Each wall S is the vertex of a scroll J(S) of lines, on which every point T represents another "companion" wall. For each companion T, its windows are centred on the same locations as those of S, but may either swell (increase in size by 2), stick (remain unchanged), or shrink (decrease by 2). Choosing a T for which a given window shrinks allows it to be perturbed in a linear fashion, in turn reducing the inductive proof mentioned earlier to elementary algebra. W. F. Lunnon "The Number-Wall Algorithm: an LFSR Cookbook" Article 01.1.1 Journal of Integer Sequences, Vol. 4 (2001) http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LUNNON/numbwall10.html Fred Lunnon [Maynooth 19/12/08]