5

$(1+ta_1)(1+ta_2)...(1+ta_n)=1$ $\forall t\in R$ if and only if $a_1=a_2=...=a_n=0$

Is it true? If yes, how to prove it?

I already gave a proof when $n=3$.

$1=(1+ta_1)(1+ta_2)(1+ta_3)=1+(a_1+a_2+a_3)t+(a_1a_2+a_1a_3+a_2a_3)t^2+(a_1a_2a_3)t^3$

It is equivalent to

$a_1+a_2+a_3=0$

$a_1a_2+a_1a_3+a_2a_3=0$

$a_1a_2a_3=0$

And this gives

$a_1=a_2=a_3=0$

Is it possible to generalize it for arbitrary $n$?

$(1+ta_1)(1+ta_2)\cdots(1+ta_n)=1$, $\forall{t}$ $\in$ $\Bbb{R}$ ⟺ $a_1=a_2=\cdots=a_n=0$

dxiv
  • 76,497
Zhenduo Cao
  • 357

2 Answers2

3

If there exist an $\,a_k \ne 0\,$ then for $\,t = -1/a_k\,$ the LHS is $\,0\,$ i.e. different from the RHS which is $\,1\,$. Therefore the equality can hold for all real $\,t\,$ iff $\,a_k = 0 \mid k=1,2,\ldots,n\,$.

Note: the above assumes that the coefficients $\,a_k\,$ are real.

dxiv
  • 76,497
  • @stressedout Added a note spelling out the underlying assumption, and also asked the OP for clarification. If the $,a_k,$ turn out to be complex, I will ammend my answer. – dxiv Sep 07 '18 at 02:28
  • 1
    I do assume ak are real numbers. Thanks. That's quite a simple way. – Zhenduo Cao Sep 07 '18 at 02:31
  • What if the numbers were complex? Not sure why it's different (just curious). – IntegrateThis Sep 07 '18 at 02:33
  • @IntegrateThis Then the proof would likely have to go along the line of the other posted answer, but it gets a bit more involved at that point. The simple argument above wouldn't work since $,t=-1/a_k,$ would not necessarily be a real number, so the given equality would not be guaranteed to hold. – dxiv Sep 07 '18 at 02:43
2

I've just found a new way to prove this.

It's not hard to see that the coefficients of polynomial are all zeros. When $n=4$, that is to say,

$a_1+a_2+a_3+a_4=0$

$a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4=0$

$a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4=0$

$a_1a_2a_3a_4=0$

By the last equality, at least one $a_i(i=1,2,3,4)$ is zero. Let's assume $a_1=0$.

Then by the second last equality, $a_2a_3a_4=0$. Let's assume $a_2=0$.

Then by the third last equality, $a_3a_4=0$. Let's assume $a_3=0$.

Then by the first equality, $a_4=0$.

Zhenduo Cao
  • 357
  • (+1) That works once you fill-in the blanks. That's essentially what (I think) this comment was hinting at. And btw, it works just as well if the $,a_k,$ are allowed to be complex. – dxiv Sep 07 '18 at 03:01
  • 1
    @dxiv: Yep, that was the gist of my comment. The polymomial $(1+ta_1)...(1+ta_n) - 1$ is zero for infinitely many $t$ (all real $t$), so must be identically 0. The coefficient on $t^n$ is the product of the $a_i$, so at least one $a_i$ must be zero. That means your really only had $n-1$ factors, so induction can take over. (And the case case is just the fact that $ta_1 = 0$ for all $t$ implies $a_1 = 0$). – Jason DeVito - on hiatus Sep 07 '18 at 03:46
  • You could use induction to prove your assumption, especially when the equation is supposed to be true for $t$ not in $\Bbb{R}$ but just for infinity many values. – dmtri Sep 07 '18 at 16:05