Polynomial methods and incidence theory / Adam Sheffer.

• Incidences and classical discrete geometry -- Basic real algebraic geometry in R² -- Polynomial partitioning -- Basic real algebraic geometry in Rd -- The joints problem and degree reduction -- Polynomial methods in finite fields -- The Elekes-Sharir-Guth-Katz framework -- Constant-degree polynomial partitioning and incidences in C² -- Lines in R³ -- Distinct distances variants -- Incidences in Rd -- Incidence applications in Rd -- Incidences in spaces over finite field -- Algebraic families, dimension counting, and ruled surfaces.

• "The past decade has seen numerous major mathematical breakthroughs for topics such as the finite field Kakeya conjecture, the cap set conjecture, Erdös's distinct distances problem, the joints problem, as well as others, thanks to the introduction of new polynomial methods. There has also been significant progress on a variety of problems from additive combinatorics, discrete geometry, and more. This book gives a detailed yet accessible introduction to these new polynomial methods and their applications, with a focus on incidence theory. Based on the author's own teaching experience, the text requires a minimal background, allowing graduate and advanced undergraduate students to get to grips with an active and exciting research front. The techniques are presented gradually and in detail, with many examples, warm-up proofs, and exercises included. An appendix provides a quick reminder of basic results and ideas"-- 

Ämnesord

Algebraisk geometri  (sao)

Combinatorial geometry.  (LCSH)

Polynomials.  (LCSH)

Incidence algebras.  (LCSH)

Partitions (Mathematics)  (LCSH)

Finite fields (Algebra)  (LCSH)

Discrete geometry.  (LCSH)

Géométrie combinatoire.  (Vedettes-matière)

Polynômes.  (Vedettes-matière)

Algèbres d'incidence.  (Vedettes-matière)

Partitions (Mathématiques)  (Vedettes-matière)

Corps finis.  (Vedettes-matière)

Géométrie discrète.  (Vedettes-matière)

Géométrie algébrique.  (Vedettes-matière)

MATHEMATICS / Discrete Mathematics.  (bisacsh)

Combinatorial geometry.  (fast)

Discrete geometry.  (fast)

Finite fields (Algebra)  (fast)

Geometry, Algebraic.  (fast)

Incidence algebras.  (fast)

Partitions (Mathematics)  (fast)

Polynomials.  (fast)

Geometry, Algebraic  (LCSH)

Klassifikation

QA167 (LCC)

511/.66 (DDC)

MAT008000 (bisacsh)

Tce (kssb/8 (machine generated))