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In an ordered field, multiplication by a positive number respects the order.

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Chapter I. Algebraic sets
Chapter II. Algebraic varieties
Chapter III. Hilbert's Nullstellensatz
Chapter IV. Real algebra

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The correspondence between algebraic sets and radical ideals

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The correspondence between algebraic sets and radical ideals

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Over an algebraically closed field, solutions of systems of polynomial equations in n variables correspond bijectively to radical ideals of the algebra of polynomials in n indeterminates.

This bijection is the basis for the equivalence of categories between affine varieties over an algebraically closed field k and finitely generated reduced k-algebras.

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The nodal cubic and its tangent cone at the origin

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The nodal cubic and its tangent cone at the origin

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The equation of the nodal cubic is y^2-x^2-x^3=0. The equation of the tangent cone is y^2-x^2=0 (the lowest-degree non-zero homogeneous term in the Taylor expansion at (0,0) of the polynomial defining the curve).

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The nodal cubic and its tangent cone at (0,0)

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The equation for this real cubic plane curve is y^2 = x(x-1)(x+1). We can use the projection to the y-axis to illustrate geometric Noether normalisation: this projection is a surjective morphism to the affine line whose fibres all contain three points (taking into account the multiplicities of such points), where 3 is the degree of the polynomial defining the curve.

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Real Algebraic Varieties - Chap. 2.1, 2.2

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Real Algebraic Varieties - Chap. 1.3, 1.4

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An equation for the Cayley cubic is (x^2-y^2)z-(x^2+y^2+z^2)+1=0.

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Real Algebraic Varieties - Chap. 1.1, 1.2