Multivariate polynomials

Now, we will consider polynomials that have more than one variable. With a single variable, everything is quite intuitive. For instance, univariate polynomials of degree $d$ have at most $d$ roots, but this is false when we have multivariate polynomials (e.g., $f(x,y) = x-y$ is a degree one polynomial but has infinitely many roots). However, you will see that we can still take advantage of the low-degree polynomials in some way and that multivariables allow you to encode much more complex problems as polynomials.

📄️ Matrix multiplication testingSingle Page

📄️ The Schwartz-Zippel LemmaSingle Page

📄️ Perfect matching detectionSingle Page

📄️ Proof of Theorem 48 (Schwartz-Zippel Lemma)Single Page