Let $p(x) = c_n x^{n} + \cdots + c_0$ be a seperable polynomial with each $c_i \in \mathbb C$, call its roots $r_i$.

The smallest gap between any two of the roots is $d = \text{min}_{i \not = j}|r_i - r_j|$.

Is there any approximate lower bound for the smallest gap? Some simple expression $F$:

$$F(c_n, \cdots, c_0) < d$$

The idea is that it should be much easier to compute than numerically approximating each root and finding the minimimum distance directly.

More generally I would like to learn about the geometry of a polynomials roots in terms of its coefficients if anyone knew some notes on books on this.


There may be a lower bound in term of the discriminant and of upper bounds $M$ for the absolute value of the roots. The discriminant $$\Delta\colon = \prod_{i < j} (x_i - x_j)^2$$ detects of all the roots are distinct and can be calculated from the coefficient of the polynomial ( a rational expression). Let $\delta$ be the smallest absolute value of a difference between roots. Then we have $$\Delta = \delta^2 \cdot \Delta'$$ where $\Delta'$ is the product of the squares of all the other differences squared. Therefore $$ |\delta|^2 = \frac{ |\Delta|} { |\Delta'|}$$ Now the denominator cannot be larger than $(2 M) ^{2(\binom{n}{2}-1)}$ and so we get a lower bound for $|\delta|$. It may well be a very crude one.


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