Edit detail for SandBoxTensorProductPolynomial revision 6 of 13

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

A tensor product is "the most general bilinear operation" available in a specified domain of computation, satisfying:

(1)

(2)

(3)

We can use the domain constructor Sum [SandBoxSum]?

(1) -> )lib SUM

   Sum is now explicitly exposed in frame initial 
   Sum will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/SUM.NRLIB/SUM

First we can define some recursive operations on the polynomials

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))

For example:

-- functions are first compiled here
--
scanPoly(x,1)

   There are 1 exposed and 3 unexposed library operations named 
      leadingMonomial having 1 argument(s) but none was determined to 
      be applicable. Use HyperDoc Browse, or issue
                         )display op leadingMonomial
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named 
      leadingMonomial with argument type(s) 
                                 Variable(x)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are 1 exposed and 3 unexposed library operations named 
      leadingMonomial having 1 argument(s) but none was determined to 
      be applicable. Use HyperDoc Browse, or issue
                         )display op leadingMonomial
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      leadingMonomial with argument type(s) 
                                 Variable(x)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

injects the polynomial x in to the tensor product. So now the full tensor product is just:

tensorPoly(p,q) == _
  scanPoly(p,1)*scanPoly(q,2)

For example:

p:=2*x^2+3

(4)

q:=5*x*y+7*y+11

(5)

r:=tensorPoly(p,q)

   >> System error:
   The function BOOT::PUT is undefined.

Demonstrating the axioms (1) (2) and (3) of the tensor product:

w:= 13*y^2+17*y+19

(6)

test( tensorPoly(p+q,w) = (tensorPoly(p,w) + tensorPoly(q,w)) )

Compiling function tensorPoly with type (Polynomial(Integer), 
      Polynomial(Integer)) -> NonNegativeInteger 

   >> System error:
   The function BOOT::|*2;scanPoly;1;initial| is undefined.

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.

)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)

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7467299293205133182-25px007.spad
      using system compiler.
   TPROD abbreviates package TensorProduct 
------------------------------------------------------------------------
   initializing NRLIB TPROD for TensorProduct 
   compiling into NRLIB TPROD 
   compiling local scanIndex : (IE,Integer) -> IEP
   Semantic Errors: 
      [1]  IEP is not a known type
      [2]  IE is not a known type

****** comp fails at level 4 with expression: ******
error in function scanIndex 

(SEQ (|:=| (|:| #1=#:G15 (|Boolean|)) (|zero?| | << x >> |))
     (|exit| 1
      (IF #1#
          0
          (+
           (|monomial| (|leadingCoefficient| |x|)
            (IF (= |n| 1)
                ((|Sel| (|Sum| VAR VAR) |in1|) (|leadingSupport| |x|))
                ((|Sel| (|Sum| VAR VAR) |in2|) (|leadingSupport| |x|))))
           (|scanIndex| (|reductum| |x|) |n|)))))
****** level 4  ******
x:= x
m:= R
f:=
((((#:G15 #) (|n| # #) (|x| # #) (|scanIndex| #) ...)))

   >> Apparent user error:
   Cannot coerce x 
      of mode IE 
      to mode (Integer)

test( p\/q = r )

   There are 2 exposed and 1 unexposed library operations named \/ 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op \/
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named \/ 
      with argument type(s) 
                             Polynomial(Integer)
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.