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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:

 (1)
 (2)
 (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)); _
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))  (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  (5) Type: Polynomial(Integer) fricas q:=5*x*y+7*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))  (7) Type: SparseMultivariatePolynomial?(Integer,Sum(Symbol,Symbol)) fricas monomials(r)  (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  (9) Type: Polynomial(Integer) fricas test( tensorPoly(p+q,w) = (tensorPoly(p,w) + tensorPoly(q,w)) )  (10) Type: Boolean fricas test( tensorPoly(p,q+w) = (tensorPoly(p,q) + tensorPoly(p,w)) )  (11) Type: Boolean fricas test( tensorPoly(p,23*w) = 23*tensorPoly(p,w) )  (12) Type: Boolean fricas test( tensorPoly(23*p,w) = 23*tensorPoly(p,w) )  (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
_\_/(p:P, q:P) : SMP == scanPoly(p,1)*scanPoly(q,2)
   Compiling FriCAS source code from file
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((\/ ((SparseMultivariatePolynomial R (Sum VAR VAR)) P P))): Not documented!!!!
; 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 )
 (14)
Type: Boolean
fricas
test( (p+q) \/ w = (p\/w) + (q\/w) )
 (15)
Type: Boolean
fricas
test( p \/ (q+w) = (p\/q) + (p\/w) )
 (16)
Type: Boolean
fricas
test( p \/ (23*w) = 23*(p\/w) )
 (17)
Type: Boolean
fricas
test( (23*p) \/ w = 23*(p\/w) )
 (18)
Type: Boolean

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

)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
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
scanPoly2(q:Q):SMP ==
q=0 => 0
_\_/(p:P, q:Q) : SMP == scanPoly1(p)*scanPoly2(q)
   Compiling FriCAS source code from file
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 )
 (19)
Type: Boolean
fricas
test( (p+q) \/ w = (p\/w) + (q\/w) )
 (20)
Type: Boolean
fricas
test( p \/ (q+w) = (p\/q) + (p\/w) )
 (21)
Type: Boolean
fricas
test( p \/ (23*w) = 23*(p\/w) )
 (22)
Type: Boolean
fricas
test( (23*p) \/ w = 23*(p\/w) )
 (23)
Type: Boolean

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

fricas
(p\/q)\/w
 (24)
Type: SparseMultivariatePolynomial?(Integer,Sum(Sum(Symbol,Symbol),Symbol))
fricas
p\/(q\/w)
 (25)
Type: SparseMultivariatePolynomial?(Integer,Sum(Symbol,Sum(Symbol,Symbol)))

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