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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
)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)
fricas
Compiling function scanIndex with type (IndexedExponents(Symbol),
      Integer) -> IndexedExponents(Sum(Symbol,Symbol))
fricas
Compiling function mapMonomial with type (Polynomial(Integer),
      Integer) -> SparseMultivariatePolynomial(Integer,Sum(Symbol,
      Symbol))
fricas
Compiling function scanPoly with type (Polynomial(Integer),Integer)
       -> SparseMultivariatePolynomial(Integer,Sum(Symbol,Symbol))
fricas
Compiling function scanPoly with type (Polynomial(Integer),Integer)
       -> SparseMultivariatePolynomial(Integer,Sum(Symbol,Symbol))
fricas
Compiling function scanPoly with type (Variable(x),Integer) -> 
      SparseMultivariatePolynomial(Integer,Sum(Symbol,Symbol))

\label{eq4}x_{1}(4)
Type: SparseMultivariatePolynomial?(Integer,Sum(Symbol,Symbol))

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{eq5}{2 \ {{x}^{2}}}+ 3(5)
Type: Polynomial(Integer)
fricas
q:=5*x*y+7*y+11

\label{eq6}{{\left({5 \  x}+ 7 \right)}\  y}+{11}(6)
Type: Polynomial(Integer)
fricas
r:=tensorPoly(p,q)
fricas
Compiling function tensorPoly with type (Polynomial(Integer),
      Polynomial(Integer)) -> SparseMultivariatePolynomial(Integer,Sum(
      Symbol,Symbol))

\label{eq7}{{\left({{\left({{10}\ {{x_{1}}^{2}}}+{15}\right)}\ {x_{2}}}+{{14}\ {{x_{1}}^{2}}}+{21}\right)}\ {y_{2}}}+{{22}\ {{x_{1}}^{2}}}+{33}(7)
Type: SparseMultivariatePolynomial?(Integer,Sum(Symbol,Symbol))
fricas
monomials(r)

\label{eq8}\left[{{10}\ {{x_{1}}^{2}}\ {x_{2}}\ {y_{2}}}, \:{{15}\ {x_{2}}\ {y_{2}}}, \:{{14}\ {{x_{1}}^{2}}\ {y_{2}}}, \:{{21}\ {y_{2}}}, \:{{2
2}\ {{x_{1}}^{2}}}, \:{33}\right](8)
Type: List(SparseMultivariatePolynomial?(Integer,Sum(Symbol,Symbol)))

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

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

\label{eq9}{{13}\ {{y}^{2}}}+{{17}\  y}+{19}(9)
Type: Polynomial(Integer)
fricas
test( tensorPoly(p+q,w) = (tensorPoly(p,w) + tensorPoly(q,w)) )

\label{eq10} \mbox{\rm true} (10)
Type: Boolean
fricas
test( tensorPoly(p,q+w) = (tensorPoly(p,q) + tensorPoly(p,w)) )

\label{eq11} \mbox{\rm true} (11)
Type: Boolean
fricas
test( tensorPoly(p,23*w) = 23*tensorPoly(p,w) )

\label{eq12} \mbox{\rm true} (12)
Type: Boolean
fricas
test( tensorPoly(23*p,w) = 23*tensorPoly(p,w) )

\label{eq13} \mbox{\rm true} (13)
Type: Boolean

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)
Time: 0.03 SEC.
compiling local mapMonomial : (P,Integer) -> SparseMultivariatePolynomial(R,Sum(VAR,VAR)) Time: 0.01 SEC.
compiling local scanPoly : (P,Integer) -> SparseMultivariatePolynomial(R,Sum(VAR,VAR)) Time: 0 SEC.
compiling exported \/ : (P,P) -> SparseMultivariatePolynomial(R,Sum(VAR,VAR)) Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |TensorProduct| REDEFINED
;;; *** |TensorProduct| REDEFINED Time: 0 SEC.
Warnings: [1] scanIndex: not known that (OrderedSet) is of mode (CATEGORY domain (IF (has VAR (Finite)) (IF (has VAR (Finite)) (ATTRIBUTE (Finite)) noBranch) noBranch) (IF (has VAR (Monoid)) (IF (has VAR (Monoid)) (ATTRIBUTE (Monoid)) noBranch) noBranch) (IF (has VAR (AbelianMonoid)) (IF (has VAR (AbelianMonoid)) (ATTRIBUTE (AbelianMonoid)) noBranch) noBranch) (IF (has VAR (CancellationAbelianMonoid)) (IF (has VAR (CancellationAbelianMonoid)) (ATTRIBUTE (CancellationAbelianMonoid)) noBranch) noBranch) (IF (has VAR (Group)) (IF (has VAR (Group)) (ATTRIBUTE (Group)) noBranch) noBranch) (IF (has VAR (AbelianGroup)) (IF (has VAR (AbelianGroup)) (ATTRIBUTE (AbelianGroup)) noBranch) noBranch) (IF (has VAR (OrderedAbelianMonoidSup)) (IF (has VAR (OrderedAbelianMonoidSup)) (ATTRIBUTE (OrderedAbelianMonoidSup)) noBranch) noBranch) (IF (has VAR (OrderedSet)) (IF (has VAR (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch) noBranch) (SIGNATURE selectsum ((Union (: acomp VAR) (: bcomp VAR)) $)) (SIGNATURE in1 ($ VAR)) (SIGNATURE in2 ($ VAR))) [2] mapMonomial: not known that (OrderedSet) is of mode (CATEGORY domain (IF (has VAR (Finite)) (IF (has VAR (Finite)) (ATTRIBUTE (Finite)) noBranch) noBranch) (IF (has VAR (Monoid)) (IF (has VAR (Monoid)) (ATTRIBUTE (Monoid)) noBranch) noBranch) (IF (has VAR (AbelianMonoid)) (IF (has VAR (AbelianMonoid)) (ATTRIBUTE (AbelianMonoid)) noBranch) noBranch) (IF (has VAR (CancellationAbelianMonoid)) (IF (has VAR (CancellationAbelianMonoid)) (ATTRIBUTE (CancellationAbelianMonoid)) noBranch) noBranch) (IF (has VAR (Group)) (IF (has VAR (Group)) (ATTRIBUTE (Group)) noBranch) noBranch) (IF (has VAR (AbelianGroup)) (IF (has VAR (AbelianGroup)) (ATTRIBUTE (AbelianGroup)) noBranch) noBranch) (IF (has VAR (OrderedAbelianMonoidSup)) (IF (has VAR (OrderedAbelianMonoidSup)) (ATTRIBUTE (OrderedAbelianMonoidSup)) noBranch) noBranch) (IF (has VAR (OrderedSet)) (IF (has VAR (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch) noBranch) (SIGNATURE selectsum ((Union (: acomp VAR) (: bcomp VAR)) $)) (SIGNATURE in1 ($ VAR)) (SIGNATURE in2 ($ VAR)))
Cumulative Statistics for Constructor TensorProduct Time: 0.04 seconds
finalizing NRLIB TPROD Processing TensorProduct for Browser database: --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/TENSOR.spad-->TensorProduct(constructor): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/TENSOR.spad-->TensorProduct((\/ ((SparseMultivariatePolynomial R (Sum VAR VAR)) P P))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/TENSOR.spad-->TensorProduct(): Missing Description ; compiling file "/var/aw/var/LatexWiki/TPROD.NRLIB/TPROD.lsp" (written 31 JUL 2013 03:47:24 PM):
; /var/aw/var/LatexWiki/TPROD.NRLIB/TPROD.fasl written ; compilation finished in 0:00:00.025 ------------------------------------------------------------------------ TensorProduct is now explicitly exposed in frame initial TensorProduct will be automatically loaded when needed from /var/aw/var/LatexWiki/TPROD.NRLIB/TPROD

fricas
test( p\/q = r )

\label{eq14} \mbox{\rm true} (14)
Type: Boolean
fricas
test( (p+q) \/ w = (p\/w) + (q\/w) )

\label{eq15} \mbox{\rm true} (15)
Type: Boolean
fricas
test( p \/ (q+w) = (p\/q) + (p\/w) )

\label{eq16} \mbox{\rm true} (16)
Type: Boolean
fricas
test( p \/ (23*w) = 23*(p\/w) )

\label{eq17} \mbox{\rm true} (17)
Type: Boolean
fricas
test( (23*p) \/ w = 23*(p\/w) )

\label{eq18} \mbox{\rm true} (18)
Type: Boolean

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)
Time: 0 SEC.
compiling local scanIndex2 : IndexedExponents VAR2 -> IndexedExponents Sum(VAR1,VAR2) Time: 0 SEC.
compiling local mapMonomial1 : P -> SparseMultivariatePolynomial(R,Sum(VAR1,VAR2)) Time: 0 SEC.
compiling local mapMonomial2 : Q -> SparseMultivariatePolynomial(R,Sum(VAR1,VAR2)) Time: 0 SEC.
compiling local scanPoly1 : P -> SparseMultivariatePolynomial(R,Sum(VAR1,VAR2)) Time: 0 SEC.
compiling local scanPoly2 : Q -> SparseMultivariatePolynomial(R,Sum(VAR1,VAR2)) Time: 0 SEC.
compiling exported \/ : (P,Q) -> SparseMultivariatePolynomial(R,Sum(VAR1,VAR2)) Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |TensorProduct2| REDEFINED
;;; *** |TensorProduct2| REDEFINED Time: 0 SEC.
Warnings: [1] scanIndex1: not known that (OrderedSet) is of mode (CATEGORY domain (IF (has VAR1 (Finite)) (IF (has VAR2 (Finite)) (ATTRIBUTE (Finite)) noBranch) noBranch) (IF (has VAR1 (Monoid)) (IF (has VAR2 (Monoid)) (ATTRIBUTE (Monoid)) noBranch) noBranch) (IF (has VAR1 (AbelianMonoid)) (IF (has VAR2 (AbelianMonoid)) (ATTRIBUTE (AbelianMonoid)) noBranch) noBranch) (IF (has VAR1 (CancellationAbelianMonoid)) (IF (has VAR2 (CancellationAbelianMonoid)) (ATTRIBUTE (CancellationAbelianMonoid)) noBranch) noBranch) (IF (has VAR1 (Group)) (IF (has VAR2 (Group)) (ATTRIBUTE (Group)) noBranch) noBranch) (IF (has VAR1 (AbelianGroup)) (IF (has VAR2 (AbelianGroup)) (ATTRIBUTE (AbelianGroup)) noBranch) noBranch) (IF (has VAR1 (OrderedAbelianMonoidSup)) (IF (has VAR2 (OrderedAbelianMonoidSup)) (ATTRIBUTE (OrderedAbelianMonoidSup)) noBranch) noBranch) (IF (has VAR1 (OrderedSet)) (IF (has VAR2 (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch) noBranch) (SIGNATURE selectsum ((Union (: acomp VAR1) (: bcomp VAR2)) $)) (SIGNATURE in1 ($ VAR1)) (SIGNATURE in2 ($ VAR2))) [2] mapMonomial1: not known that (OrderedSet) is of mode (CATEGORY domain (IF (has VAR1 (Finite)) (IF (has VAR2 (Finite)) (ATTRIBUTE (Finite)) noBranch) noBranch) (IF (has VAR1 (Monoid)) (IF (has VAR2 (Monoid)) (ATTRIBUTE (Monoid)) noBranch) noBranch) (IF (has VAR1 (AbelianMonoid)) (IF (has VAR2 (AbelianMonoid)) (ATTRIBUTE (AbelianMonoid)) noBranch) noBranch) (IF (has VAR1 (CancellationAbelianMonoid)) (IF (has VAR2 (CancellationAbelianMonoid)) (ATTRIBUTE (CancellationAbelianMonoid)) noBranch) noBranch) (IF (has VAR1 (Group)) (IF (has VAR2 (Group)) (ATTRIBUTE (Group)) noBranch) noBranch) (IF (has VAR1 (AbelianGroup)) (IF (has VAR2 (AbelianGroup)) (ATTRIBUTE (AbelianGroup)) noBranch) noBranch) (IF (has VAR1 (OrderedAbelianMonoidSup)) (IF (has VAR2 (OrderedAbelianMonoidSup)) (ATTRIBUTE (OrderedAbelianMonoidSup)) noBranch) noBranch) (IF (has VAR1 (OrderedSet)) (IF (has VAR2 (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch) noBranch) (SIGNATURE selectsum ((Union (: acomp VAR1) (: bcomp VAR2)) $)) (SIGNATURE in1 ($ VAR1)) (SIGNATURE in2 ($ VAR2)))
Cumulative Statistics for Constructor TensorProduct2 Time: 0 seconds
finalizing NRLIB TPROD2 Processing TensorProduct2 for Browser database: --->-->TensorProduct2(constructor): Not documented!!!! --->-->TensorProduct2((\/ ((SparseMultivariatePolynomial R (Sum VAR1 VAR2)) P Q))): Not documented!!!! --->-->TensorProduct2(): Missing Description ; compiling file "/var/aw/var/LatexWiki/TPROD2.NRLIB/TPROD2.lsp" (written 31 JUL 2013 03:47:24 PM):
; /var/aw/var/LatexWiki/TPROD2.NRLIB/TPROD2.fasl written ; compilation finished in 0:00:00.035 ------------------------------------------------------------------------ TensorProduct2 is now explicitly exposed in frame initial TensorProduct2 will be automatically loaded when needed from /var/aw/var/LatexWiki/TPROD2.NRLIB/TPROD2

fricas
test( p\/q = r )

\label{eq19} \mbox{\rm true} (19)
Type: Boolean
fricas
test( (p+q) \/ w = (p\/w) + (q\/w) )

\label{eq20} \mbox{\rm true} (20)
Type: Boolean
fricas
test( p \/ (q+w) = (p\/q) + (p\/w) )

\label{eq21} \mbox{\rm true} (21)
Type: Boolean
fricas
test( p \/ (23*w) = 23*(p\/w) )

\label{eq22} \mbox{\rm true} (22)
Type: Boolean
fricas
test( (23*p) \/ w = 23*(p\/w) )

\label{eq23} \mbox{\rm true} (23)
Type: Boolean

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

fricas
(p\/q)\/w

\label{eq24}\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\left({{\left({{130}\ {{{x_{1}}_{1}}^{2}}}+{195}\right)}\ {{x_{2}}_{1}}}+{{182}\ {{{x_{1}}_{1}}^{2}}}+{273}\right)}\ {{y_{2}}_{1}}}+ 
\
\
\displaystyle
{{286}\ {{{x_{1}}_{1}}^{2}}}+{429}
(24)
Type: SparseMultivariatePolynomial?(Integer,Sum(Sum(Symbol,Symbol),Symbol))
fricas
p\/(q\/w)

\label{eq25}\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\left({{\left({{130}\ {{x_{1}}^{2}}}+{195}\right)}\ {{x_{1}}_{2}}}+{{182}\ {{x_{1}}^{2}}}+{273}\right)}\ {{y_{1}}_{2}}}+ 
\
\
\displaystyle
{{286}\ {{x_{1}}^{2}}}+{429}
(25)
Type: SparseMultivariatePolynomial?(Integer,Sum(Symbol,Sum(Symbol,Symbol)))




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