* compute partials
* jacobian; eval @ pt.
* f2applot?

* columnar fpn / fpnpm output
* get deg vs. find deg
* replace some polrat stuff with ED stuff

* ED stuff into regression scripts
* declare default fpnpoly_t & then set_coeff.  this examines coeff[0]
  which is still set to default invalid; abend.  to fix, have set_coeff
  recognize default (invalid) member data.

----------------------------------------------------------------
* virtuals ... code duplication is getting to be a pain. 
* fp->fq embed for tmvpoly into utils func
* tmvpoly zero poly: include a zero monom w/ correct nvars?

* separate homogenize & not.  remove duplicated code.
  - agff homogenize
  - zeroset -h
  - singular points -h: show all evals, show zeroes, show just count
  - partials -h
  - eval
  k agff clump by conjugates
* genus (using Pluecker formula)
* list intersection points (needs tarray class ...)
k points of degree n ...
* intersection divisors
* function divisor: div(C, f) w/ f = g/h.  div(f) = sum P - sum Q,
  where P = C \cap C_g and Q = C \cap C_h (intersection divisors).
* tmvrat class, w/ scan poly/poly
* Fq(C): equivalence test: g/h ~ g'/h' iff gh' - g'h in <F>
  Use Groebner for ideal-membership test?

----------------------------------------------------------------
* other 2 glist functions

* correct totients for double extensions when inner modulus is reducible.
  Or caveat emptor.
! memoize lots of functions -- phi? gcd?
* fppm log tbl.  needs find_gen & what else?

k bit_matrix_t to accelerate f2pfactor

! other row-red routine for rings (must be EDs).  Needs generic "gcd".

* skip-whitespace blob into linescan.

----------------------------------------------------------------
k check that each spiff cmd calls the correct usage func.  OR, make it
  right: put usage func (perh. null) into lookup tbl.
----------------------------------------------------------------
* post some tables
----------------------------------------------------------------
* conversion:

.   q f2 f2n fp fpn
q   - .  .   .  .
f2  - -  .   .  .
f2n - .  -   .  .
fp  - .  .   -  .
fpn - .  .   .  -

----------------------------------------------------------------
* separate out the split-poly function
* spiff f*split
* eigenvalues
* eigenvectors
* min pol?
* jordan form?
* rational canonical form?

- get matrix
- form chpol
- factor chpol
- find lcm of degrees of factors
- find irr for splitting extension (search or cpstm?)
  >> embed, or extend?
- split chpol over that splitting field; get eigenvalues
- for each eigenvalue,
  - compute nullity of A - t I
  - if nullity != multiplicity, not diag'ble

- for each eigenvalue,
  - compute kerbas
  - populate D and P matrices
- have automated checker

= mx to chpol (have)
= chpol -> splitter modulus and array of eigenvalues
= splitter mod & eigenvalues -> diagonalizability test
= kerbas (have)

----------------------------------------------------------------
* fp_find_gen to intmod_find_gen
* modexp method for polys; into p-berl.
* are iss's really ever needed?  whack 'em.
* fppoly_random to use set_coeff now.
* 1x1 ~ scalar in tmat stuff, for benefit of cmd_line_mat_parse.
* DEBUG -> VERBOSE

----------------------------------------------------------------
* chpol:
  > f2   -> f2pr
  > fp   -> fppr
  > f2pm -> f2npr
  > fppm -> fpnpr

* compmx: do remaining classes

----------------------------------------------------------------
* crt template
  - vector of moduli
  - vector of residues
  - vector of mod_t

* f2px, fppx utils for dir prod, and routines for conversion.
  > istream -- ?  do I need a new class after all, or just some utilsp routines?

* pluck method for *rat_t, and into det callers.

----------------------------------------------------------------
k special bit matrix
* sighdlr in find/rand irrs for status "wiid"
* rename *linalg.* to something more appropriate.
* fix tfac binary search on f2pfactor 1e

----------------------------------------------------------------
! regression tests:
  - rand det w/ check
  - rand inv w/ check
  - rand factor w/ check
  - auto-test square roots in all poly factor?

----------------------------------------------------------------
* FF mat ord must be a multiple of ord of det.  Code it up.
k f2ipm order finder (in C, as well) for non-irr, using CRT + cache?
k z*list

----------------------------------------------------------------
? fe list: all, units, generator, order=spec
? re list: all, monic, irred, prim, show period, show factors, classify

! primitivity testing for all
! clean up usages
! check iss.fail() in app mains etc.
* ffcomp: alg'ms, pfs out of appces & into own sections.
k scalar * vec op*?
k vec * matrix op*?
k bitmatrix to speed up f2pfactor

================================================================
for ffcomp:

if h^q equiv h mod f, then f = prod c in Fq gcd(f, h-c)
pf: Easily, RHS divides LHS.  h-c's are rel prime so prod c in Fq gcd(f,h-c)
is (?) gcd(f, prod(h-c)).  Now, prod c in Fq (h-c) is h^q - h (pv!).  Since
this is equiv 0 mod f, done.

How to find such f-reducers?  Use lin alg.
h(x)=sum i=0 n-1 a_i x^i
h(x)^q = h(x^q) = sum i=0 n-1 a_i x^{iq}
w/r/t s.b., (a_{n-1} .. a_0) = B (a_{n-1} - a_0)
Solve for nullspace of (B-I)(a_{n-1}, ..., a_0)

What is B?  Show by example.

5
.  n=5
^4+1 (7 * b).  n=5
x^2 = 0015+x^4+1 (7 * b).  n=5
x^4 = 10000
x^6 = 10011
x^8 = 11101

a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 x
= a_4(x^4+x^3+x^2+1)
+ a_3(x^4+x+1)
+ a_2(x^4)
+ a_1(x^2)
+ a_0(1)

[ a_4 ]
[ a_3 ]
[ a_2 ]
[ a_1 ]
[ a_0 ]

B
[ 1 1 1 0 0 ]
[ 1 0 0 0 0 ]
[ 1 0 0 1 0 ]
[ 1 1 0 0 0 ]
[ 1 1 0 0 1 ]

(n-1-j)th column is x^{jq} mod f

B-I
[ 0 1 1 0 0 ]
[ 1 1 0 0 0 ]
[ 1 0 1 1 0 ]
[ 1 1 0 1 0 ]
[ 1 1 0 0 0 ]

1 0 1 0 0 (row echelon)
0 1 1 0 0
0 0 0 1 0
0 0 0 0 0
0 0 0 0 0

1 1 1 0 0 (kerbas)
0 0 0 0 1

check 1c^2 equiv 1c mod 31

... etc.