Ways to protect certain parts of expressions

Question

I am currently working on a quantum mechanics problem where I try to find equations that relate different states of a many-body system. The states are described by sets of pairs of integers, like list1 = {{0,0},{0,1},{1,-1},{3,-3}}. There is a very expensive and complicated function listToPoly[li_List] that turn such lists into polynomials. It turns out that many seemingly different lists can produce the same polynomial, or that sets of lists produce sets of polynomials that are linearly dependent. I have found a way to determine some of these dependencies without using listToPoly but I would like some help implementing it.

What I want to accomplish is a way to represent these lists in a way that I can both act on them with functions that manipulate their list structure, e.g.,

func1[li_List,k_] := Transpose[{#1+k, #2}]& @@ Transpose[li]

and programmatically generate and solve the equations that represent the linear dependences. Say I have three lists:

l1 = {{0, 0}, {0, 1}}
l2 = {{0, 0}, {0, 2}}
l3 = {{0, 1}, {0, 2}}

and I find out that their corresponding polynomials p1, p2, p3 would satisfy

2*p1 - 3*p2 + p3 == 0

I would like to be able to generate such equations using the lists (I don't want to invoke listToPoly), but then I need to prevent evaluation like

2*l1 - 3*l2 + l3 = {{0, 1}, {0, -2}}

Even combining the two things, it would be nice to be able to define something like

lindep[li_List, kmax_] := Sum[(-1)^k * func1[li, k], {k, kmax}] == 0

and then evaluate

lindep[{{0,0},{0,1}}, 2]

to get

- {{1,0},{1,1}} + {{2,0},{2,1}} == 0

There are probably lots of ways to do this, like:

• Using Hold and ReleaseHold
• Adding or replacing a custom head and defining functions to act on object with that head
• Using ToString and ToExpression back and forth
• etc.

Question

What would you guys choose? Any clear (dis)advantages to any particular method? I realize I am basically asking "how to handle custom objects", but I thought giving the context would make it easier to point me in the right direction.

Answer 1

I realize I am basically asking "how to handle custom objects", but I thought giving the context would make it easier to point me in the right direction.

It seems that you are. I believe the most natural way to do that in Mathematica is to use a custom head. I'll use obj for my examples.

First you might define a pattern for your custom object:

p0 = obj[{{_, _} ..}];

Then define a new func1 (I'll call fn1) referencing that pattern:

fn1[li : p0, k_] := MapAt[# + k &, li, {1, All, 1}]

And lindep:

lindep[li : p0, kmax_] := Sum[(-1)^k*fn1[li, k], {k, kmax}] == 0

We can define a Format to style obj expressions as plain lists:

Format[x : p0] := Interpretation[First @ x, x]

Finally:

lindep[obj[{{0, 0}, {0, 1}}], 2]

-{{1, 0}, {1, 1}} + {{2, 0}, {2, 1}} == 0