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However large $n$ may be, the equation

$x_1^2 + \ldots + x_n^2 = -1$ has no real solution.

(Similarly, the homogeneous equation $x_1^2 + \ldots + x_n^2 = 0$ has no nontrivial solution.)

If you want to go there, the theorem which gives you a necessary and sufficient condition for a system of real polynomial equations to have a real solution is the Real Nullstellensatz. See for instance Section 3.9` of

http://www.math.uga.edu/~pete/modeltheory2010Chapter3.pdfhttp://alpha.math.uga.edu/~pete/modeltheory2010Chapter3.pdf

(Very roughly, this theorem says that the above is essentially the only way that a system of $n$ real polynomial equations in more than $n$ variables can fail to have a real solution.)

However large $n$ may be, the equation

$x_1^2 + \ldots + x_n^2 = -1$ has no real solution.

(Similarly, the homogeneous equation $x_1^2 + \ldots + x_n^2 = 0$ has no nontrivial solution.)

If you want to go there, the theorem which gives you a necessary and sufficient condition for a system of real polynomial equations to have a real solution is the Real Nullstellensatz. See for instance Section 3.9` of

http://www.math.uga.edu/~pete/modeltheory2010Chapter3.pdf

(Very roughly, this theorem says that the above is essentially the only way that a system of $n$ real polynomial equations in more than $n$ variables can fail to have a real solution.)

However large $n$ may be, the equation

$x_1^2 + \ldots + x_n^2 = -1$ has no real solution.

(Similarly, the homogeneous equation $x_1^2 + \ldots + x_n^2 = 0$ has no nontrivial solution.)

If you want to go there, the theorem which gives you a necessary and sufficient condition for a system of real polynomial equations to have a real solution is the Real Nullstellensatz. See for instance Section 3.9` of

http://alpha.math.uga.edu/~pete/modeltheory2010Chapter3.pdf

(Very roughly, this theorem says that the above is essentially the only way that a system of $n$ real polynomial equations in more than $n$ variables can fail to have a real solution.)

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Pete L. Clark
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However large $n$ may be, the equation

$x_1^2 + \ldots + x_n^2 = -1$ has no real solution.

(Similarly, the homogeneous equation $x_1^2 + \ldots + x_n^2 = 0$ has no nontrivial solution.)

If you want to go there, the theorem which gives you a necessary and sufficient condition for a system of real polynomial equations to have a real solution is the Real Nullstellensatz. See for instance Section 3.9` of

http://www.math.uga.edu/~pete/modeltheory2010Chapter3.pdf

(Very roughly, this theorem says that the above is essentially the only way that a system of $n$ real polynomial equations in more than $n$ variables can fail to have a real solution.)

However large $n$ may be, the equation

$x_1^2 + \ldots + x_n^2 = -1$ has no real solution.

(Similarly, the homogeneous equation $x_1^2 + \ldots + x_n^2 = 0$ has no nontrivial solution.)

If you want to go there, the theorem which gives you a necessary and sufficient condition for a system of real polynomial equations to have a real solution is the Real Nullstellensatz. See for instance Section 3.9` of

http://www.math.uga.edu/~pete/modeltheory2010Chapter3.pdf

However large $n$ may be, the equation

$x_1^2 + \ldots + x_n^2 = -1$ has no real solution.

(Similarly, the homogeneous equation $x_1^2 + \ldots + x_n^2 = 0$ has no nontrivial solution.)

If you want to go there, the theorem which gives you a necessary and sufficient condition for a system of real polynomial equations to have a real solution is the Real Nullstellensatz. See for instance Section 3.9` of

http://www.math.uga.edu/~pete/modeltheory2010Chapter3.pdf

(Very roughly, this theorem says that the above is essentially the only way that a system of $n$ real polynomial equations in more than $n$ variables can fail to have a real solution.)

Source Link
Pete L. Clark
  • 65.4k
  • 12
  • 241
  • 381

However large $n$ may be, the equation

$x_1^2 + \ldots + x_n^2 = -1$ has no real solution.

(Similarly, the homogeneous equation $x_1^2 + \ldots + x_n^2 = 0$ has no nontrivial solution.)

If you want to go there, the theorem which gives you a necessary and sufficient condition for a system of real polynomial equations to have a real solution is the Real Nullstellensatz. See for instance Section 3.9` of

http://www.math.uga.edu/~pete/modeltheory2010Chapter3.pdf