A Fundamental Theorem of Algebra type Result

Asked Sep 11 '20 at 00:23

Active Sep 11 '20 at 02:33

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Consider a polynomial $$p(t)=\sum_{i=1}^n a_it^{b_i}+a_0$$ of degree $b_n$, show that it admits at most $n+1$ nonnegative roots and at least one complex root. Assume all $b_i$ are positive and $a_i$ are real numbers.

Attempt: Using successively Rolle's Theorem we can prove that it admits at most $2^{n-1}$ nonnegative roots.

edited Sep 11 '20 at 02:33

asked Sep 11 '20 at 00:23

checkmath

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What does $[b_n]$ mean ? – Simon Sep 11 '20 at 00:34
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The integer part of $b_n$. – checkmath Sep 11 '20 at 00:37
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Is this an exercise or a conjecture ? – Simon Sep 11 '20 at 00:39
Something in between. – checkmath Sep 11 '20 at 00:42
https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs – Qiaochu Yuan Sep 11 '20 at 01:43
Are the $a_n$ real or complex? – mjw Sep 11 '20 at 02:01
@QiaochuYuan, I'm pretty sure the OP means for the exponents $b_i$ to be positive reals, not necessarily integers. Does the rule of signs generalize? – Barry Cipra Sep 11 '20 at 13:10

A Fundamental Theorem of Algebra type Result

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