Content (Tentative)

• A quick introduction to notions of rings, fields and other prelims
• Euclidean algorithm for GCD of univariate polynomials, Resultant and their properties
• Fast Fourier transform, polynomial division, univariate multipoint evaluation
• Univariate polynomial factorization over finite fields: Cantor-Zassenhaus and Berlekemp's algorithms
• Bivariate polynomial factorization, Hensel lifting and Newton iteration
• Multivariate polynomial factorization: Kaltofen's algorithms
• Hilbert's irreducibility theorem
• Factorization of univariate polynomials over rationals, LLL Basis reduction algorithm
• Primality testing: randomized and deterministic algorithms
• Multivariate multipoint evaluation over finite fields and a data structure for polynomial evaluation
• Ideals, varieties and ideal membership problem
• Groebner basis and application to Ideal membership
• Emptiness of varieties and weak form of Hilbert's Nullstellansatz
• Strong form of Hilbert's Nullstellansatz, Quantifier elimination
• Algebraic independence, the Jacobian criterion and testing algebraic independence, applications
• Bezout's theorem, Wooley's theorem and applications
• Algebraic models of computation, Ben Or and Cleve, Strassen's degree bound, Baur and Strassen
• Algebraic P vs NP and connections to its more famous Boolean Cousin
• More things if time permits...