FindInstance returns Indeterminate in version 9, but not in 8

Question

Bug introduced in 9.0 and fixed in 10.0.0

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Here is a trivial system of equations in three unknowns for which FindInstance obtains a solution:

FindInstance[
 c3 + c2 == 1 && c1 == 0 && -c2 == 0, {c1, c2, c3}]

(* ==> {{c1 -> 0, c2 -> 0, c3 -> 1}} *)

But in my application, I have a linear system where I want to rule out the trivial solution c1==c2==c3 with an inequality:

FindInstance[
 (c3 != 0 || c2 != 0) && c1 == 0 && -c2 == 0, {c1, c2, c3}]

(* ==> {{c1 -> 0, c2 -> Indeterminate, c3 -> 1}} *)

This incorrectly returns Indeterminate for c2 only in Mathematica version 9, whereas it gives a correct result c2 -> 0 in version 8. In general, my variables are complex so that I can't restrict the domain to Reals (which solves the problem in this simple example).

My current solution to this issue is to replace the inequality by an equation, e.g., Norm[{c1, c2, c3}] ==1. But I would like to understand (if at all possible) what change between versions is causing the Indeterminate result. I have a version-8 notebook that suddenly produced Indeterminate in over half its calculations, and there is no documented change in FindInstance that I'm aware of. So I'd like to hear what the safest and/or most efficient way would be to rule out the trivial solution in FindInstance for a system of linear equations.

Answer 1

This problem can be fixed with the following patch (you can put it in your init.m file).

Unprotect[Internal`FromCoefficientList];
Internal`FromCoefficientList[cfs_List, 0] := If[cfs==={}, 0, cfs[[1]]];
Protect[Internal`FromCoefficientList];

Resolving the problem:

FindInstance[(c3 != 0 || c2 != 0) && c1 == 0 && -c2 == 0, {c1, c2, c3}]
(* {{c1 -> 0, c2 -> 0, c3 -> 1}} *)

Thanks for pointing it out.

Answer 2

On v10.0.0,

In[1]:= FindInstance[(c3 != 0 || c2 != 0) && c1 == 0 && -c2 == 0, {c1, c2, c3}]

Out[1]= {{c1 -> 0, c2 -> 0, c3 -> 1}}