# Trinomials with given roots

## TL;DR

This paper investigates the number of trinomials that vanish at specified complex numbers, establishing bounds on their quantity, degrees, and heights, especially in the algebraic case, with some exceptions.

## Contribution

The paper provides new bounds on the number, degrees, and heights of trinomials with given roots, extending previous results and including effective bounds in the algebraic case.

## Key findings

- Number of such trinomials is bounded by an absolute constant
- Effective bounds on degrees and heights for algebraic roots
- Identifies exceptions where bounds do not apply

## Abstract

We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these trinomials.