Real algebraic geometry

Real algebraic geometry is a fun part of mathematics where we study shapes and spaces. We do this by solving equations with real numbers. It is a part of something bigger called algebraic geometry, which looks at how algebra and geometry connect. In real algebraic geometry, we focus on finding answers that are real numbers and see how they relate.

A related area is semialgebraic geometry. This looks at solving not just equations, but also inequalities with real numbers. These answers form groups called semialgebraic sets. Mathematicians study how these sets can be changed using special rules called semialgebraic mappings.

These ideas help us understand complicated shapes and spaces. They are useful in many areas like computer graphics, making things better (optimization), and physics. By studying real answers to math problems, mathematicians can solve real-world problems.

Terminology

In real algebraic geometry, we study shapes and patterns made by real numbers that solve equations. These shapes are called real algebraic sets.

Sometimes, when we look at these shapes from different angles, they might not stay as real algebraic sets, but they still fit into a broader category called semialgebraic sets.

There are many related areas of study, like o-minimal theory and real analytic geometry. Real plane curves and polyhedra are examples of these sets. Computational real algebraic geometry focuses on using computer algorithms to understand and work with these shapes. Real algebra looks at special number systems and how they relate to these geometric shapes.

Main article: Tarski–Seidenberg theorem
Main articles: Real plane curves, polyhedra
Main article: cylindrical algebraic decomposition
Main articles: ordered fields, ordered rings, real closed fields, positive polynomials, sums-of-squares of polynomials
Main articles: Hilbert's 17th problem, Krivine's Positivestellensatz
Main articles: commutative algebra, complex algebraic geometry
Main articles: moment problems, convex optimization, quadratic forms, valuation theory, model theory

Timeline of real algebra and real algebraic geometry

Real algebraic geometry is a part of mathematics. It looks at how to solve equations using real numbers. This timeline shows some important moments.

Key moments include Fourier's work in 1826 on solving groups of inequalities. In 1835, Sturm's theorem showed how to count real roots. In 1876, Harnack's curve theorem was introduced. In 1900, Hilbert's problems helped guide math for much of the 1900s. In 1931, Alfred Tarski created a new way to solve some math problems. This was later improved by Abraham Seidenberg. Over the years, many mathematicians worked on understanding the shapes and features of solutions to algebraic equations. This led to important findings, like John Nash's discovery in 1952 that every smooth shape can be shown using algebra.