3

Let $S = (P_1,P_2, \ldots,P_n)$ be a set of polynomials with complex coefficients.

I call S critical if set of solutions of $\{P_n(a) = P_n(b)\forall n \}$ in $\Bbb C^2$ is $\{ (a=b) \cup (\text{some finite points in }\Bbb C^2)\}$ I call a set minimal critical set if it is a critical set but no proper subset of it is critical.

Examples of minimal critical sets are (a*z +b) ($z^2, z^3$) or ($z^2 +z, z^4$)

Question is does there exist minimal critical sets of cardinality n for every n ? for example what are the examples for n=3,4 with proof?

It is easy to find infinite subsets T of polynomials which do not have any minimal critical subset, for example $\{P(z^2)\ \forall P\}$, but then is there a condition on such sets?

Leo
  • 7,670
  • 5
  • 31
  • 63
rohit
  • 963

1 Answers1

2

Let $m=2\cdot 3\cdot 5\cdots p_n$ be the product of the first $n$ primes. Let $P_k(x)=x^{m/p_k}$. Then $P_k(a)=P_k(b)$ implies $|a|=|b|$ and (assuming they are nonzero) $\zeta:=\frac ab$ is a $\frac m{p_k}$th root of unity, that is the order of $\zeta$ is a divisor of $\frac m{p_k}$. Thus if $P_k(a)=P_k(b)$ for all $k$, we conclude that the order of $\zeta$ is $1$, i.e. $a=b$. Hence the set is critical.

On the other hand, if just one $k$ is missing, then $b=\xi a$ with $\xi$ a nontrivial $p_k$th root of unity gives you infinitely many solutions, i.e. no proper subset of the above is critical.

Hagen von Eitzen
  • 374,180
  • if i put one additional condition on Pi(z) that each Pi(z) is not of the form (z-a)^k +c for any a,k,c , then how to tackle? – rohit May 11 '13 at 14:13