# Is there a word for the degree of how much a system of equations is underdetermined?

Is there a word for the degree of how much a system of equations is underdetermined?

Does the word

underdetermanancy

exist? I also found

underdetermination

but it seems to be a philosophical term.

Background: The values of a system of (say) 100 variables is in general determined by at least 100 equations. If you just have 99 equations the system is slightly underdetermined. If you just have a few equations its highly underdetermined.

Edit

In other words: Is there a word for the ratio (number of equations)/(number of variables)?

• The noun formed from indeterminate is indeterminacy, so a (much rarer) noun formed from underdeterminate is underdeterminacy (no n before the c). Sep 12, 2016 at 15:44

I would say that a system of equation with what you're calling "underdeterminancy k" has

k degrees of freedom.

If the values of all the variables are determined, a system has no degrees of freedom. If you have 100 variables and 99 equations, you generally have one degree of freedom. And if you have just a few equations, it has many degrees of freedom.

• Actually, I would like to write something general, without the precise knowlegde of k. Like "sparse signals are guaranteed to be recovered perfectly depending on the underdetermanancy of the system". The ratio (number of equations)/(number of variables) is important in these contexts.
– Rob
Sep 12, 2016 at 11:57

The word you are looking for is indeterminate. [Wikipedia]

In mathematics:

Indeterminate (variable), a symbol that is treated as a variable

Indeterminate system, a system of simultaneous equations that has more than one solution

Indeterminate equation, an equation that has more than one solution

Indeterminate form, an algebraic expression with certain limiting behaviour in mathematical analysis

The noun is indeterminancy: the condition or quality of being indeterminate. See Dictionary.com, for example.

• No, thats different. Indeterminancy means that there is something unknown before you solve for it. An underdetermined system just means, that there are not enough equations to solve it the classical way. Further a system can either be indetermined or not. There is nothing in between.
– Rob
Sep 12, 2016 at 11:38
• @Rob First, near the end of your question, I assume you meant to say "99 equations", not "99 variables". Okay, I get it. You're looking for a word for the degree to which a system is underdetermined. I should have seen. In that case, I would start start with something like "degrees of freedom", as suggested by Peter Shor. A linear system with 99 equations and 100 variables has 1 degree of freedom; a linear system with 5 equations and 100 variables has 95 degrees of freedom. Sep 12, 2016 at 13:09
• I agree that degree of freedom is acceptable, but often, you want to maintain, or emphasize, the sort of distinctions you just discussed above, because in a well posed linear analysis, these distinctions can can save you a paragraph of explanations further down the road. Degree of freedom is a hypernym of this specific case. I'd stick with degree of underdeterminedness when talking about the system of equations themselves. Once you have crunched them, then you can switch to degree of freedom to characterize the results, because all the distinctions have been lost to the crunching process. Sep 12, 2016 at 15:14
• Degree of both is a count, not a ratio. I haven't ever seen a term for the ratio of equations to unknowns. You should include that point in your question. Sep 12, 2016 at 15:22
• @Rob I support your use of a ratio to characterize the degree. It's a form of normalization. I would consider inverting the ratio though, i.e., using (# variables)/(# equations), so that as the ratio goes up, the degree of underdeterminedness goes up. So in your example we would have the following possible degrees, ranging from 1 to 100: 100/100=1, 100/99, ... 100/2, 100/1=100. Something like that anyway. Sep 12, 2016 at 16:06