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SandBoxTensorProductPolynomial

Edit detail for SandBoxTensorProductPolynomial revision 13 of 13

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Editor: test1 Time: 2026/05/30 23:48:40 GMT+0
Note:

changed:
-scanPoly(x,1)
scanPoly(x+1,1)

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+1,1)

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

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

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

Type: Void

For example:

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p:=2*x^2+3

$\label{eq4}{2 \ {{x}^{2}}}+ 3$

(4)

Type: Polynomial(Integer)

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

$\label{eq5}{{\left({5 \ x}+ 7 \right)}\ y}+{11}$

(5)

Type: Polynomial(Integer)

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r:=tensorPoly(p,q)

fricas

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

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

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

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

$\label{eq6}{{13}\ {{y}^{2}}}+{{17}\ y}+{19}$

(6)

Type: Polynomial(Integer)

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

   >> 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 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: 0.01 SEC.

   compiling local mapMonomial : (P,Integer) -> SparseMultivariatePolynomial(R,Sum(VAR,VAR))
Time: 0 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)))
      [3] scanPoly: 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: 0.02 seconds

   finalizing NRLIB TPROD 
   Processing TensorProduct for Browser database:
--->/usr/local/lib/fricas/target/x86_64-linux-gnu/../../src/algebra/TENSOR.spad-->TensorProduct(constructor): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-linux-gnu/../../src/algebra/TENSOR.spad-->TensorProduct((\/ ((SparseMultivariatePolynomial R (Sum VAR VAR)) P P))): Not documented!!!!
--->/usr/local/lib/fricas/target/x86_64-linux-gnu/../../src/algebra/TENSOR.spad-->TensorProduct: Missing Description
; compiling file "/var/aw/var/LatexWiki/TPROD.NRLIB/TPROD.lsp" (written 22 OCT 2026 07:04:27 PM):

; wrote /var/aw/var/LatexWiki/TPROD.NRLIB/TPROD.fasl
; compilation finished in 0:00:00.020
------------------------------------------------------------------------
   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 )

   >> System error:
   The function BOOT::|Sum| is undefined.

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 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)
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)))
      [3] scanPoly1: 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)))
      [4] scanPoly2: 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: 0.01 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 22 OCT 2026 07:04:27 PM):

; wrote /var/aw/var/LatexWiki/TPROD2.NRLIB/TPROD2.fasl
; compilation finished in 0:00:00.036
------------------------------------------------------------------------
   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 )

   >> System error:
   The function BOOT::|Sum| is undefined.

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

fricas

(p\/q)\/w

   >> System error:
   The function BOOT::|Sum| is undefined.