page_title: Algebraic geometry
section_title: Zeros of simultaneous polynomials
hierarchical_section_title: Algebraic geometry / Basic notions / Zeros of simultaneous polynomials
caption_attribution_description: English: Mathematica commands: In[132]:= &lt;&lt;Graphics`Shapes` In[133]:= Solve[{x^2+y^2+z^2\[Equal]1,x+y+z\[Equal]0},{x,y}] Out[133]= \!\({{x \[Rule] 1\/2\ \((\(-z\) - \@\(2 - 3\ z\^2\))\), y \[Rule] 1\/2\ \((\(-z\) + \@\(2 - 3\ z\^2\))\)}, {x \[Rule] 1\/2\ \((\(-z\) + \@\(2 - 3\ z\^2\))\), y \[Rule] 1\/2\ \((\(-z\) - \@\(2 - 3\ z\^2\))\)}}\) In[134]:= \!\(\(p1[z_] := {1\/2\ \((\(-z\) - \@\(2 - 3\ z\^2\))\), 1\/2\ \((\(-z\) + \@\(2 - 3\ z\^2\))\), z};\)\[IndentingNewLine] \(p2[z_] := {1\/2\ \((\(-z\) + \@\(2 - 3\ z\^2\))\), 1\/2\ \((\(-z\) - \@\(2 - 3\ z\^2\))\), z};\)\[IndentingNewLine] \) In[136]:= \!\(\(Show[{\[IndentingNewLine]WireFrame[ Graphics3D[ Sphere[1, 20, 20]]], \[IndentingNewLine]Graphics3D[{\[IndentingNewLine]\ RGBColor[1, 0, 0], \[IndentingNewLine]Thickness[ 0.009], \[IndentingNewLine]Line[ Table[p1[z], {z, \(-\@\(2\/3\)\), \@\(2\/3\), 0.01}]], \[IndentingNewLine]Line[ Table[p2[z], {z, \(-\@\(2\/3\)\), \@\(2\/3\), 0.01}]], \[IndentingNewLine]Text[ StyleForm["\&lt;x+y+z==0\&gt;", Section, FontColor \[Rule] RGBColor[1, 0, 0]], {1, 0.2, 1}, {\(-1\), 0}]}]\[IndentingNewLine]}, Boxed \[Rule] False, Axes \[Rule] True, ViewPoint -&gt; {8.043, \ \(-2.956\), \ 1.784}, Boxed \[Rule] True, AxesEdge \[Rule] {{1, \(-1\)}, {1, \(-1\)}, {1, 1}}, AxesLabel \[Rule] {StyleForm[x, Section], StyleForm[y, Section], StyleForm[z, Section]}, Ticks \[Rule] {{\(-1\), \(- .5\), 0, .5, 1}, {\(-1\), \(- .5\), 0, .5, 1}, {\(-1\), \(- .5\), 0, .5, 1}}, PlotLabel \[Rule] StyleForm[TraditionalForm[x^2 + y^2 + z^2 \[Equal] 1]\ , Section, FontColor -&gt; RGBColor[0, 0, 0]]]\ ;\)\)
context_page_description: Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
  The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
context_section_description: In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R³ could be defined as the set of all points (x,y,z) with
  A "slanted" circle in R³ can be defined as the set of all points (x,y,z) which satisfy the two polynomial equations
caption: Sphere and slanted circle
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