Edit detail for SandBoxTensorProductPolynomial revision 9 of 12

http://en.wikipedia.org/wiki/Tensor_product

A tensor product is "the most general bilinear operation" available in
a specified domain of computation, satisfying:
\begin{equation}
\label{eq1}
(v_1+v_2)\otimes w - (v_1\otimes w+v_2\otimes w) = 0
\end{equation}
\begin{equation}
\label{eq2}
v\otimes (w_1+w_2) - (v\otimes w_1+v\otimes w_2) = 0
\end{equation}
\begin{equation}
\label{eq3}
cv\otimes w=v\otimes cw=c(v\otimes w)
\end{equation}
We can use the domain constructor Sum [SandBoxSum]
\begin{axiom}
)lib SUM
\end{axiom}
First we can define some recursive operations on the polynomials
\begin{axiom}
scanPoly(p,n) == _
  (p=0 => 0; mapMonomial(leadingMonomial(p),n)+scanPoly(reductum p,n))
mapMonomial(p,n) == 
  monomial(coefficient(p,degree p),scanIndex(degree(p),n))$SMP(Integer,Sum(Symbol,Symbol))
scanIndex(p,n) == 
  (zero? p => 0$IndexedExponents(Sum(Symbol,Symbol)); 
    monomial(leadingCoefficient(p), 
      if n=1 then in1(leadingSupport(p))$Sum(Symbol,Symbol) 
             else in2(leadingSupport(p))$Sum(Symbol,Symbol) 
    )$IndexedExponents(Sum(Symbol,Symbol))+ _
      scanIndex(reductum(p),n))
\end{axiom}
For example:
\begin{axiom}
-- functions are first compiled here
--
scanPoly(x,1)
\end{axiom}
injects the polynomial x in to the tensor product. So
now the full tensor product is just:
\begin{axiom}
tensorPoly(p,q) == _
  scanPoly(p,1)*scanPoly(q,2)
\end{axiom}
For example:
\begin{axiom}
p:=2x^2+3
q:=5xy+7y+11
r:=tensorPoly(p,q)
monomials(r)
\end{axiom}
Demonstrating the axioms (1) (2) and (3) of the tensor product:
\begin{axiom}
w:= 13y^2+17y+19
test( tensorPoly(p+q,w) = (tensorPoly(p,w) + tensorPoly(q,w)) )
test( tensorPoly(p,q+w) = (tensorPoly(p,q) + tensorPoly(p,w)) )
test( tensorPoly(p,23w) = 23tensorPoly(p,w) )
test( tensorPoly(23p,w) = 23tensorPoly(p,w) )
\end{axiom}
I suppose that we could give an inductive proof that this
implementation of the tensor product of polynomials is
correct ... but for now lets take this demonstration as
reassurance.
Re-coding the interpreter functions as library package.
\begin{spad}
)abbrev package TPROD TensorProduct
macro IE == IndexedExponents(VAR)
macro IEP == IndexedExponents(Sum(VAR,VAR))
macro SMP == SparseMultivariatePolynomial(R,Sum(VAR,VAR))
TensorProduct(R:Ring, VAR: OrderedSet, P:PolynomialCategory(R,IE,VAR)): with
    \/: (P,P) -> SMP
  == add
    scanIndex(x:IE,n:Integer):IEP ==
      zero? x => 0
      monomial(leadingCoefficient(x), 
        if n=1 then in1(leadingSupport(x))$Sum(VAR,VAR) 
               else in2(leadingSupport(x))$Sum(VAR,VAR) _
      ) + scanIndex(reductum(x),n)
    mapMonomial(p:P,n:Integer):SMP ==
      monomial(coefficient(p,degree p),scanIndex(degree(p),n))$SMP
    scanPoly(p:P,n:Integer):SMP ==
      p=0 => 0
      mapMonomial(leadingMonomial(p),n)+scanPoly(reductum p,n)
    (p:P \/ q:P):SMP == scanPoly(p,1)*scanPoly(q,2)
\end{spad}

\begin{axiom}
test( p\/q = r )
test( (p+q) \/ w = (p\/w) + (q\/w) )
test( p \/ (q+w) = (p\/q) + (p\/w) )
test( p \/ (23w) = 23(p\/w) )
test( (23p) \/ w = 23(p\/w) )
\end{axiom}
Here's another way to write this - maybe better this way as first
step to express associativity of the tensor product.
\begin{spad}
)abbrev package TPROD2 TensorProduct2
macro IE1 == IndexedExponents(VAR1)
macro IE2 == IndexedExponents(VAR2)
macro S == Sum(VAR1,VAR2)
macro IEP == IndexedExponents(S)
macro SMP == SparseMultivariatePolynomial(R,S)
TensorProduct2(R:Ring, VAR1: OrderedSet, VAR2: OrderedSet, P:PolynomialCategory(R,IE1,VAR1), Q:PolynomialCategory(R,IE2,VAR2)): with
    \/: (P,Q) -> SMP
  == add
    scanIndex1(x:IE1):IEP ==
      zero? x => 0
      monomial(leadingCoefficient(x), in1(leadingSupport(x))$S) + scanIndex1(reductum(x))
    scanIndex2(x:IE2):IEP ==
      zero? x => 0
      monomial(leadingCoefficient(x), in2(leadingSupport(x))$S) + scanIndex2(reductum(x))
    mapMonomial1(p:P):SMP ==
      monomial(coefficient(p,degree p),scanIndex1(degree(p)))$SMP
    mapMonomial2(q:Q):SMP ==
      monomial(coefficient(q,degree q),scanIndex2(degree(q)))$SMP
    scanPoly1(p:P):SMP ==
      p=0 => 0
      mapMonomial1(leadingMonomial(p))+scanPoly1(reductum p)
    scanPoly2(q:Q):SMP ==
      q=0 => 0
      mapMonomial2(leadingMonomial(q))+scanPoly2(reductum q)
    (p:P \/ q:Q):SMP == scanPoly1(p)*scanPoly2(q)
\end{spad}

\begin{axiom}
test( p\/q = r )
test( (p+q) \/ w = (p\/w) + (q\/w) )
test( p \/ (q+w) = (p\/q) + (p\/w) )
test( p \/ (23w) = 23(p\/w) )
test( (23p) \/ w = 23(p\/w) )
\end{axiom}
Associativity of the tensor product means these two expressions
should be identical:
\begin{axiom}
(p\/q)\/w
p\/(q\/w)
\end{axiom}


    Some or all expressions may not have rendered properly,
    because Axiom returned the following error:
Error: export AXIOM=/usr/local/lib/open-axiom/x86_64-unknown-linux/1.3.0-2009-01-05; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 240; $AXIOM/bin/AXIOMsys < /var/zope2/var/LatexWiki/6578669110478838235-25px.axm
Segmentation fault

GCL (GNU Common Lisp)  2.6.7 CLtL1    Oct 29 2006 20:31:33
Source License: LGPL(gcl,gmp), GPL(unexec,bfd,xgcl)
Binary License:  GPL due to GPL'ed components: (XGCL READLINE BFD UNEXEC)
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Use (help) to get some basic information on how to use GCL.
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             OpenAxiom: The Open Scientific Computation Platform 
                     Version: OpenAxiom 1.3.0-2009-01-05
               Built on Wednesday January 7, 2009 at 17:48:30 
-----------------------------------------------------------------------------
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(1) -> (1) -> (1) -> (1) -> (1) -> )lib SUM
   Sum is now explicitly exposed in frame initial 
   Sum will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/SUM.NRLIB/code.o
(1) -> scanPoly(p,n) == _
  (p=0 => 0; mapMonomial(leadingMonomial(p),n)+scanPoly(reductum p,n))
                                                                   Type: Void
mapMonomial(p,n) == _
  monomial(coefficient(p,degree p),scanIndex(degree(p),n))$SMP(Integer,Sum(Symbol,Symbol))
                                                                   Type: Void
scanIndex(p,n) == 
  (zero? p => 0$IndexedExponents(Sum(Symbol,Symbol)); 
    monomial(leadingCoefficient(p), 
      if n=1 then in1(leadingSupport(p))$Sum(Symbol,Symbol) 
             else in2(leadingSupport(p))$Sum(Symbol,Symbol) 
    )$IndexedExponents(Sum(Symbol,Symbol))+ 
      scanIndex(reductum(p),n))
                                                                   Type: Void
(4) -> -- functions are first compiled here
--
scanPoly(x,1)

! Missing $ inserted.
<inserted text> 
                $
l.120     
           \/: (P,P) -> SMP
 Missing $ inserted.
<inserted text> 
                $
l.125 ...hen in1(leadingSupport(x))$Sum(VAR,VAR) _
 Missing $ inserted.
<inserted text> 
                $
l.126 ...lse in2(leadingSupport(x))$Sum(VAR,VAR) _

Overfull \hbox (127.90825pt too wide) in paragraph at lines 119--133
\OML/cmm/m/it/12 Integer\OT1/cmr/m/n/12 ) : \OML/cmm/m/it/12 IEP \OT1/cmr/m/n/1
2 == \OML/cmm/m/it/12 zero\OT1/cmr/m/n/12 ?\OML/cmm/m/it/12 x \OT1/cmr/m/n/12 =
\OML/cmm/m/it/12 > \OT1/cmr/m/n/12 0\OML/cmm/m/it/12 monomial\OT1/cmr/m/n/12 (\
OML/cmm/m/it/12 leadingCoefficient\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m
/n/12 )\OML/cmm/m/it/12 ;[] fn \OT1/cmr/m/n/12 = 1\OML/cmm/m/it/12 thenin\OT1/c
mr/m/n/12 1(\OML/cmm/m/it/12 leadingSupport\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\
OT1/cmr/m/n/12 ))$Sum(VAR,VAR) $[]\OML/cmm/m/it/12 lsein\OT1/cmr/m/n/12 2(\OML/
cmm/m/it/12 leadingSupport\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 ))
$Sum(VAR,VAR)
Overfull \hbox (8.49167pt too wide) in paragraph at lines 119--133
[] \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 scanIndex\OT1/cmr/m/n/12 (\OML/cmm/m/it/1
2 reductum\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 
; n\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 mapMonomial\OT1/cmr/m/n/12 (\OML/cmm/m/it/
12 p \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 P; n \OT1/cmr/m/n/12 : \OML/cmm/m/it/12
 Integer\OT1/cmr/m/n/12 ) : \OML/cmm/m/it/12 SMP \OT1/cmr/m/n/12 == \OML/cmm/m/
it/12 monomial\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 coefficient\OT1/cmr/m/n/12 (\OM
L/cmm/m/it/12 p; degreep\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 ; scanIndex\OT1/cmr/m
/n/12 (\OML/cmm/m/it/12 degree\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 p\OT1/cmr/m/n/1
2 )\OML/cmm/m/it/12 ; n\OT1/cmr/m/n/12 ))$SMP
[4]
LaTeX Warning: Characters dropped after `\end{axiom}' on input line 142.
 Missing $ inserted.
<inserted text> 
                $
l.152     
           \/: (P,Q) -> SMP
 Missing $ inserted.
<inserted text> 
                $
l.170 
Overfull \hbox (22.17155pt too wide) in paragraph at lines 151--170
[]\OT1/cmr/m/n/12 TensorProduct2(R:Ring, VAR1: Or-dered-Set, VAR2: Or-dered-Set
, P:PolynomialCategory(R,IE1,VAR1), Q:PolynomialCategory(R,IE2,VAR2)):
Overfull \hbox (0.20291pt too wide) in paragraph at lines 151--170
\OT1/cmr/m/n/12 with $[]\OML/cmm/m/it/12 = \OT1/cmr/m/n/12 : (\OML/cmm/m/it/12 
P; Q\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > SMP \OT1/cmr/m/n/1
2 == \OML/cmm/m/it/12 addscanIndex\OT1/cmr/m/n/12 1(\OML/cmm/m/it/12 x \OT1/cmr
/m/n/12 : \OML/cmm/m/it/12 IE\OT1/cmr/m/n/12 1) : \OML/cmm/m/it/12 IEP \OT1/cmr
/m/n/12 == \OML/cmm/m/it/12 zero\OT1/cmr/m/n/12 ?\OML/cmm/m/it/12 x \OT1/cmr/m/
n/12 =\OML/cmm/m/it/12 > \OT1/cmr/m/n/12 0\OML/cmm/m/it/12 monomial\OT1/cmr/m/n
/12 (\OML/cmm/m/it/12 leadingCoefficient\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1
/cmr/m/n/12 )\OML/cmm/m/it/12 ; in\OT1/cmr/m/n/12 1(\OML/cmm/m/it/12 leadingSup
port\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 ))$S)
Overfull \hbox (171.6808pt too wide) in paragraph at lines 151--170
\OT1/cmr/m/n/12 + scanIn-dex1(reductum(x)) scanIn-dex2(x:IE2):IEP == zero?