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  "body": "\n## Summary\n\nReal algebraic geometry studies the sets of real solutions of systems of polynomial equations and inequalities, and the polynomials that are nonnegative on them.\nIts basic objects are real algebraic sets and [[semialgebraic sets|BX5WSK8ZCT]] over the [[real numbers|08J2M1JYAG]] or over a [[real closed field|DDY3Q3PJ6W]].\nIts main tools are the [[Tarski–Seidenberg theorem|3W5JGDZAN0]], orderings of fields, and certificates of positivity such as the [[Positivstellensatz|FN734RQYM6]].\nIt has applications to optimization, robotics and the topology of real curves and surfaces.\n\n## Explanations\n\n### L1 human\n\nOver the complex numbers every nonconstant polynomial in one variable has a root, and algebraic geometry is clean.\nOver the real numbers, $x^2 + 1$ has no root, and inequalities like $x^2 + y^2 \\le 1$ matter as much as equations.\nReal algebraic geometry studies such real shapes: discs, ovals, regions bounded by polynomial curves.\nA typical question asks how many pieces such a shape can have.\nAnother asks how to prove that a polynomial never takes negative values, for instance by writing it as a sum of squares.\n",
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