---
tag: PPWJ8NFH7G
kind: concept
name: Real algebraic geometry
msc: [14P05, 14P10, "12D15"]
sources:
  - url: https://en.wikipedia.org/wiki/Real_algebraic_geometry
same_as:
  wikidata: Q7301119
  wikipedia: Real algebraic geometry
---

## Summary

Real algebraic geometry studies the sets of real solutions of systems of polynomial equations and inequalities, and the polynomials that are nonnegative on them.
Its basic objects are real algebraic sets and [[semialgebraic sets|BX5WSK8ZCT]] over the [[real numbers|08J2M1JYAG]] or over a [[real closed field|DDY3Q3PJ6W]].
Its main tools are the [[Tarski–Seidenberg theorem|3W5JGDZAN0]], orderings of fields, and certificates of positivity such as the [[Positivstellensatz|FN734RQYM6]].
It has applications to optimization, robotics and the topology of real curves and surfaces.

## Explanations

### L1 human

Over the complex numbers every nonconstant polynomial in one variable has a root, and algebraic geometry is clean.
Over the real numbers, $x^2 + 1$ has no root, and inequalities like $x^2 + y^2 \le 1$ matter as much as equations.
Real algebraic geometry studies such real shapes: discs, ovals, regions bounded by polynomial curves.
A typical question asks how many pieces such a shape can have.
Another asks how to prove that a polynomial never takes negative values, for instance by writing it as a sum of squares.