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KConrad
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See the accepted answer at https://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theorem for a multivariable Hensel's lemma for n$n$ polynomials in n$n$ variables (more general versions are possible). Or see https://kconrad.math.uconn.edu/blurbs/gradnumthy/multivarhensel.pdf.

It is not a consequence of the classical one-variable case for one polynomial, just as linear algebra in arbitrary dimensions is not a special case of one dimension. But with the right concepts and notation (e.g., Jacobians) the proof is similar to the one-dimensional case.

See the accepted answer at https://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theorem for a multivariable Hensel's lemma for n polynomials in n variables (more general versions are possible).

It is not a consequence of the classical one-variable case for one polynomial, just as linear algebra in arbitrary dimensions is not a special case of one dimension. But with the right concepts and notation (e.g., Jacobians) the proof is similar to the one-dimensional case.

See the accepted answer at https://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theorem for Hensel's lemma for $n$ polynomials in $n$ variables (more general versions are possible). Or see https://kconrad.math.uconn.edu/blurbs/gradnumthy/multivarhensel.pdf.

It is not a consequence of the classical one-variable case for one polynomial, just as linear algebra in arbitrary dimensions is not a special case of one dimension. But with the right concepts and notation (e.g., Jacobians) the proof is similar to the one-dimensional case.

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See the accepted answer at http://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theoremhttps://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theorem for a multivariable Hensel's lemma for n polynomials in n variables (more general versions are possible).

It is not a consequence of the classical one-variable case for one polynomial, just as linear algebra in arbitrary dimensions is not a special case of one dimension. But with the right concepts and notation (e.g., Jacobians) the proof is similar to the one-dimensional case.

See the accepted answer at http://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theorem for a multivariable Hensel's lemma for n polynomials in n variables (more general versions are possible).

It is not a consequence of the classical one-variable case for one polynomial, just as linear algebra in arbitrary dimensions is not a special case of one dimension. But with the right concepts and notation (e.g., Jacobians) the proof is similar to the one-dimensional case.

See the accepted answer at https://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theorem for a multivariable Hensel's lemma for n polynomials in n variables (more general versions are possible).

It is not a consequence of the classical one-variable case for one polynomial, just as linear algebra in arbitrary dimensions is not a special case of one dimension. But with the right concepts and notation (e.g., Jacobians) the proof is similar to the one-dimensional case.

Source Link
KConrad
KConrad
  • 50.6k
  • 9
  • 196
  • 277

See the accepted answer at http://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theorem for a multivariable Hensel's lemma for n polynomials in n variables (more general versions are possible).

It is not a consequence of the classical one-variable case for one polynomial, just as linear algebra in arbitrary dimensions is not a special case of one dimension. But with the right concepts and notation (e.g., Jacobians) the proof is similar to the one-dimensional case.