### MS-E1142 - Computational Algebraic Geometry D, 11.01.2021-19.02.2021

Please note! Course description is confirmed for two academic years (1.8.2018-31.7.2020), which means that in general, e.g. Learning outcomes, assessment methods and key content stays unchanged. However, via course syllabus, it is possible to specify or change the course execution in each realization of the course, such as how the contact sessions are organized, assessment methods weighted or materials used.

#### LEARNING OUTCOMES

The student will learn how to solve systems of polynomial equations and describe images of polynomial maps. The student will understand the correspondence between affine varieties (solution sets of polynomial equations) on the geometry side and ideals on the algebra side. The student will learn how to do computations using Groebner bases and the theory behind the computations. This course can be seen as a nonlinear extension of linear algebra.

Credits: 5

Schedule: 11.01.2021 - 19.02.2021

Teacher in charge (valid 01.08.2020-31.07.2022): Kaie Kubjas

Teacher in charge (applies in this implementation): Kaie Kubjas

Contact information for the course (applies in this implementation):

CEFR level (applies in this implementation):

Language of instruction and studies (valid 01.08.2020-31.07.2022):

Teaching language: English

Languages of study attainment: English

##### Content
• Valid 01.08.2020-31.07.2022:

You will learn the definitions of an affine variety and an ideal together with examples, basic properties and the correspondence between ideals and varieties. You will familiarize yourself with the method of Groebner basis which allows to study ideals computationally. You will learn how to eliminate variables from systems of polynomial equations, and how this is applied to solving systems of polynomial equations and describing images of polynomial maps. You will see an application of the theory.

##### Assessment Methods and Criteria
• Valid 01.08.2020-31.07.2022:

Teaching methods: lectures and exercises.

Assessment methods: exercises and a final exam.

• Valid 01.08.2020-31.07.2022:

Contact hours 36h (no compulsory attendance), self-study ca 100h.

#### DETAILS

##### Study Material
• Valid 01.08.2020-31.07.2022:

"Ideals, Varieties and Algorithms" by Cox, Little and O Shea

##### Prerequisites
• Valid 01.08.2020-31.07.2022:

MS-C134X Lineaarialgebra / Linear algebra. MS-C1081 Abstract algebra is recommended.