# A short proof for the parameter continuation theorem

## TL;DR

This paper provides a concise proof of the parameter continuation theorem, fundamental in polynomial homotopy continuation, using Gr"obner bases, and introduces a method for computing discriminants.

## Contribution

It offers a new, shorter proof of the parameter continuation theorem and a novel approach to computing discriminants via Gr"obner bases.

## Key findings

- Short proof of the parameter continuation theorem
- Method for computing discriminants using Gr"obner bases
- Enhanced understanding of polynomial homotopy continuation

## Abstract

The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gr\"obner bases. Our approach gives a method for computing discriminants.