Expository paper from a directed study in algebraic geometry. The resultant allow us to check if the polynomials $f$ and $g$ share a root just using their coefficients. Using these ideas, we can prove a more general theorem in algebraic geometry called the fundamental theorem of elimination theory.
Abstract. Swan’s theorem establishes a module structure on the set of smooth sections over a vector bundle $\pi\colon E\to M$.
This allows us to study algebra by means
of vector bundles and vice-versa. We conclude with an introductory result in K-theory that follows from this theorem
Abstract. This is an expository paper on the prerequisites for Gårding’s inequality.
Proposed by Lars Gårding, this inequality has applications in the
study of weak solutions to elliptic partial differential equations.
We will give the prerequisites to
state Gårding’s inequality and give one application.
MATH 475: Introduction to combinatorics
Notes (Chapter 14: Burnside’s Lemma and Polya Counting) (PDF)