NCIndependenceCheck[aListofLists,variable]

Aliases: None.

Description: NCIndependenceCheck[aListofLists,variable] is aimed at verifying whether or not a given set of polynomials are independent or not. It analyzes each list of polynomials in aListofLists separately. There are three possible types of outputs for each list in aListofLists. Two of them correspond to NCIndependenceCheck successfully determining whether or not the list of polynomials is independent. The third type of output corresponds to an unsuccessful attempt at determining dependence or independence. If a particular list is determined to be independent, True will be returned. If a list is determined to be dependent, a list beginning with False containing a set of coefficients which demonstrate independence will be returned. Finally, if NCIndependenceCheck cannot determine dependence or independence, it returns a list beginning with Undetermined containing other information which is illustrated below and described further in Comments/Limitations.

Arguments: aListofLists is a list containing a list of the polynomials which are suspected of being dependent. The argument variable will be subscripted and used to return the coefficient dependencies for each list. Below is an example of a list of four lists. The first two are dependent, the third is independent, and the fourth is undetermined.


Suppose you have four lists:

$$List1 = \{7,6a,a,abd^2,d,b,12a,d,4a^2d,a^2,5a^2,b^2,b \}$$

$$List2 = \{50,8a,a,abd^2,d,b,12a,d,4a^2d,a^2,16a^2,40b^2,b \}$$

$$List3 = \{4a,5b + c,c \}$$

$$List4 = \{x**y,y**x\}$$

Then NCIndependenceCheck[List1,List2,List3,List4,$\lambda$] returns
$\{NewList1,NewList2,NewList3,NewList4 \}$ where:

$$NewList1 = \{False,$$

$$\{0 ,-\frac {\lambda_3}{6} - 2\lambda_7, \lambda_3,0, -\lambda_8,... ...mbda_7,\lambda_8,-\frac {5\lambda_{11}}{4}, 0,\lambda_{11},0,\lambda_{13} \} \}$$

$$NewList2 = \{False,$$

$$\{0 , -\frac {\lambda_3}{8} - \frac {3\lambda_7}{2}, \lambda_3,0,... ...da_{13},\lambda_7,\lambda_8,-4\lambda_{11}, 0,\lambda_{11},0,\lambda_{13} \} \}$$

$$NewList3 = True$$

$$NewList4 = \{Undetermined , -\lambda_2x**y+\lambda_2y**x,\{-\lambda_2, \lambda_2 \} \}$$

In particular, what the above says is that $List1.Newlist1[[2]] = 0$,
and $List2.Newlist2[[2]] = 0$ (where ``.'' refers to the vector dot product). Therefore, the set of polynomials in $List1$ and $List2$ are dependent. $List3$ is independent. Note that $List4$ is clearly indpendent in the noncommutating case, and dependent in the commuting case. When such phenomena occur, NCIndependenceCheck is unable to determine whether or not the list of polynomials is independent. However, it does return to the user, a set of dependencies for the $\lambda_i$'s which must hold in order for the polynomials to sum to zero.

Comments / Limitations: IndependenceCheck first uses the CommuteEverything command to make the problem feasible. Therefore it is possible that polynomials are dependent if variables commute, and independent if not. So in this case, or when the the expression does not collapse to zero when using the commuting coefficients with the non commuting polynomials, then the list $\{Undertermined, expression ,list\}$ is returned. The list element expression is the sum of the polynomials with their corresponding $\lambda$'s. And finally, list yields a list of the dependencies for the coefficents.