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Bug introduced in 9.0 and fixed in 10.0.0

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

 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:

 (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.

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Here's a workaround: For some reason, Mathematica only returns the Indeterminate if you ask for 1 result. Ask for 2 and take the first one. – Xerxes Mar 7 '13 at 22:36
@Xerxes That's funny. But then my workaround with Norm still gives a simpler result. – Jens Mar 7 '13 at 23:38
I will second Adam's "thanks for pointing it out". – Daniel Lichtblau Mar 8 '13 at 0:21
Sorry for taking the version-9 off, but I thought that was the policy adopted in this meta discussion. – Michael E2 Aug 14 '15 at 19:50
@MichaelE2 Oh, I wasn't aware of that, or maybe forgot. It's OK - I'll remove the tag again. Too bad we don't have a real bug tracker. – Jens Aug 14 '15 at 20:31
up vote 19 down vote accepted

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

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

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.

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I tried it on about twenty different systems of equations, and everything is back to what it looked like in version 8. So the problem seems to be solved. Thanks for the quick response! – Jens Mar 7 '13 at 23:45
Welcome to Mathematica.StackExchange, Adam. – Mr.Wizard Mar 8 '13 at 4:42
Maybe it would be better to surround this with If[$Version == 9, ...] to avoid trouble after any future upgrades (since users will likely keep their init.m but might forget about modifications). – Szabolcs Dec 6 '13 at 18:51

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}}
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