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I am considering the problem of determining the algebraic independence of $n$ polynomials in $m$ variables with real coefficients, where $m \geq n$. The variables will be denoted by $a_{1}, a_{2}, ... , a_{m}$. For example, consider the following example system,

$f_{1} = a_{3}a_{4}$

$f_{2} = a_{3}a_{5}$

$f_{3} = a_{7}(a_{3}+a_{6})$

$f_{4} = a_{5}(a_{2}+a_{4})$

$f_{5} = -a_{1}(a_{2}+a_{4})$

$f_{6} = -a_{1}a_{5}$

Any system I am considering is in general of this form - i.e., sums of products of two of the possible variables. I believe in this case it is easy to see that these are dependent - for example, set $f_{6} = 0$, then $a_{1} = 0$ or $a_{5} = 0$. In either of these cases, other polynomials must be 0 as well.

However, I have been dealing with other systems in which it is not so easy to determine. Is there any way to do so in general, or any tricks I could use? For example, I was considering taking the log of the polynomials, proving linear independence of the resulting equations, and then using the Lindemann-Weierstrass theorem to state algebraic independence of the polynomials. However, the $a_{i}$ may be 0 or negative, so I'm not convinced this is general enough. Any help would be appreciated.

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    $\begingroup$ Perhaps this will help you: mathoverflow.net/questions/41535/… $\endgroup$
    – J.C. Ottem
    Mar 6, 2013 at 23:53
  • $\begingroup$ Ah,thank you. I looked around for a similar question but did not find one - perhaps I did not look hard enough. $\endgroup$
    – mnh1364
    Mar 6, 2013 at 23:53
  • $\begingroup$ As a follow up question, can these polynomials be algebraically independent, but the algebraic equations themselves not be? For example, I am looking at these equations as polynomials in the $a_{i}$ because it seems advantageous, but they may more accurately be described simply as algebraic equations - i.e. this really is a system of equations. $\endgroup$
    – mnh1364
    Mar 7, 2013 at 0:20
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    $\begingroup$ hmm, what do you mean by dependency of algebraic equations, as opposed to polynomials? $\endgroup$
    – Dima Pasechnik
    Mar 7, 2013 at 3:19
  • $\begingroup$ Similar question recently posted to m.se, math.stackexchange.com/questions/323075/… $\endgroup$
    – Gerry Myerson
    Mar 7, 2013 at 4:29

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