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

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


\label{eq1}
(v_1+v_2)\otimes w - (v_1\otimes w+v_2\otimes w) = 0
(1)

\label{eq2}
v\otimes (w_1+w_2) - (v\otimes w_1+v\otimes w_2) = 0
(2)

\label{eq3}
cv\otimes w=v\otimes cw=c(v\otimes w)
(3)

We can use the domain constructor Sum SandBoxSum

fricas
(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

fricas
scanPoly(p,n) == _
  (p=0 => 0; mapMonomial(leadingMonomial(p),n)+scanPoly(reductum p,n))
Type: Void
fricas
mapMonomial(p,n) == _
  monomial(coefficient(p,degree p),scanIndex(degree(p),n))$SMP(Integer,Sum(Symbol,Symbol))
Type: Void
fricas
scanIndex(p,n) == _
  (zero? p => 0$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

For example:

fricas
-- 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:

fricas
tensorPoly(p,q) == _
  scanPoly(p,1)*scanPoly(q,2)
Type: Void

For example:

fricas
p:=2*x^2+3

\label{eq4}{2 \ {{x}^{2}}}+ 3(4)
Type: Polynomial(Integer)
fricas
q:=5*x*y+7*y+11

\label{eq5}{{\left({5 \  x}+ 7 \right)}\  y}+{11}(5)
Type: Polynomial(Integer)
fricas
r:=tensorPoly(p,q)
>> System error: #<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/SUM.NRLIB/SUM.fasl" {10021C7463}> is a fasl file compiled with SBCL 1.1.1, and can't be loaded into SBCL 2.2.9.debian.

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

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

\label{eq6}{{13}\ {{y}^{2}}}+{{17}\  y}+{19}(6)
Type: Polynomial(Integer)
fricas
test( tensorPoly(p+q,w) = (tensorPoly(p,w) + tensorPoly(q,w)) )
fricas
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.

spad
)abbrev package TPROD TensorProduct
IE ==> IndexedExponents(VAR)
IEP ==> IndexedExponents(Sum(VAR,VAR))
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)
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6318234874982058351-25px007.spad
      using old system compiler.
   TPROD abbreviates package TensorProduct 
------------------------------------------------------------------------
   initializing NRLIB TPROD for TensorProduct 
   compiling into NRLIB TPROD 
   compiling local scanIndex : (IndexedExponents VAR,Integer) -> IndexedExponents Sum(VAR,VAR)
****** comp fails at level 7 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 7 ****** $x:= ((Sel (Sum VAR VAR) in1) (leadingSupport x)) $m:= VAR $f:= ((((#:G15 # #) (|n| # #) (|x| # #) (|scanIndex| #) ...)))
>> Apparent user error: Cannot coerce x of mode (IndexedExponents VAR) to mode ##3

fricas
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.

Here's another way to write this - maybe better this way as first step to express associativity of the tensor product.

spad
)abbrev package TPROD2 TensorProduct2
IE1 ==> IndexedExponents(VAR1)
IE2 ==> IndexedExponents(VAR2)
S ==> Sum(VAR1,VAR2)
IEP ==> IndexedExponents(S)
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)
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4565958775469301848-25px009.spad
      using old system compiler.
   TPROD2 abbreviates package TensorProduct2 
------------------------------------------------------------------------
   initializing NRLIB TPROD2 for TensorProduct2 
   compiling into NRLIB TPROD2 
   compiling local scanIndex1 : IndexedExponents VAR1 -> IndexedExponents Sum(VAR1,VAR2)
****** comp fails at level 6 with expression: ******
error in function scanIndex1 
(SEQ (|:=| (|:| #1=#:G16 (|Boolean|)) (|zero?| |x|)) (|exit| 1 (IF #1# 0 (+ (|monomial| (|leadingCoefficient| |x|) | << | ((|Sel| (|Sum| VAR1 VAR2) |in1|) (|leadingSupport| |x|)) | >> |) (|scanIndex1| (|reductum| |x|)))))) ****** level 6 ****** $x:= ((Sel (Sum VAR1 VAR2) in1) (leadingSupport x)) $m:= VAR1 $f:= ((((#:G16 # #) (|x| # . #1=#) (|scanIndex1| #) (|x| . #1#) ...)))
>> Apparent user error: Cannot coerce x of mode (IndexedExponents VAR1) to mode ##5

fricas
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.

Associativity of the tensor product means these two expressions should be identical:

fricas
(p\/q)\/w
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.




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