$\DeclareMathOperator{\pp}{\mathbb{P}}$My aim here is to have a collection of "natural" not-so-common reformulations/extensions of common/well-known results such that

1. the reformulation/extension should provide/facilitate new insights, and
2. it should not require concepts which are more "technical" than the original/common formulation.

For example, a (the most?) common formulation of Bézout's theorem is the following:

The number (counted with multiplicity) of zeroes on $\pp^n$ of $n$ homogeneous polynomials in $n+1$ variables (over an algebraically closed field) is the product of their degrees provided the number of solutions is finite.

Expressed this way, Bézout's theorem says nothing when the number of zeroes is infinite, and on this site there are questions caused by the resulting incomplete understanding. A more natural formulation, which gives additional information at zero additional cost is the following:

The number (counted with multiplicity) of isolated zeroes on $\pp^n$ of $n$ homogeneous polynomials in $n+1$ variables (over an algebraically closed field) is at most the product of their degrees. This bound is achieved if the number of zeroes is finite.

The more elaborate, but in a sense more elementary, "affine" version of Bézout's theorem gives more information on the number of zeroes on the affine space, again requiring no new concepts (although it does remove the information about the number of zeroes on the projective space):

The number (counted with multiplicity) of isolated zeroes of $n$ polynomials in $n$ variables (over an algebraically closed field) is at most the product of their degrees. This bound is achieved if and only if the only common zero of the leading forms of these polynomials is the origin.

Similar observations apply to common formulations of other "Bézout-type" theorems, e.g. the Bernstein-Kushnirenko theorem. In fact a formulation of the latter theorem in a recently posted expository article on arxiv is what prompted this question. I might add it as an answer later if no one takes on the mantle.

$\DeclareMathOperator{\pp}{\mathbb{P}}$My aim here is to have a collection of "natural" not-so-common reformulations/extensions of common/well-known results such that

1. the reformulation/extension should provide/facilitate new insights, and
2. it should not require concepts which are more "technical" than the original/common formulation.

For example, a (the most?) common formulation of Bézout's theorem is the following:

The number (counted with multiplicity) of zeroes on $\pp^n$ of $n$ homogeneous polynomials in $n+1$ variables (over an algebraically closed field) is the product of their degrees provided the number of solutions is finite.

Expressed this way, Bézout's theorem says nothing when the number of zeroes is infinite, and on this site there are questions caused by the resulting incomplete understanding. A more natural formulation, which gives additional information at zero additional cost is the following:

The number (counted with multiplicity) of isolated zeroes on $\pp^n$ of $n$ homogeneous polynomials in $n+1$ variables (over an algebraically closed field) is at most the product of their degrees. This bound is achieved if and only if the number of zeroes on $\pp^n$ is finite.

The more elaborate, but in a sense more elementary, "affine" version of Bézout's theorem gives more information on the number of zeroes on the affine space, again requiring no new concepts (although it does remove the information about the number of zeroes on the projective space):

The number (counted with multiplicity) of isolated zeroes of $n$ polynomials in $n$ variables (over an algebraically closed field) is at most the product of their degrees. This bound is achieved if and only if the only common zero of the leading forms of these polynomials is the origin.

Similar observations apply to common formulations of other "Bézout-type" theorems, e.g. the Bernstein-Kushnirenko theorem. In fact a formulation of the latter theorem in a recently posted expository article on arxiv is what prompted this question. I might add it as an answer later if no one takes on the mantle.

$\DeclareMathOperator{\pp}{\mathbb{P}}$My aim here is to have a collection of "natural" not-so-common reformulations/extensions of common/well-known results such that

1. the reformulation/extension should provide/facilitate new insights, and
2. it should not require concepts which are more "technical" than the original/common formulation.

For example, a (the most?) common formulation of Bézout's theorem is the following:

The number (counted with multiplicity) of zeroes on $\pp^n$ of $n$ homogeneous polynomials in $n+1$ variables (over an algebraically closed field) is the product of their degrees provided the number of solutions is finite.

Expressed this way, Bézout's theorem says nothing when the number of zeroes is infinite, and on this site there are questions caused by the resulting incomplete understanding. A more natural formulation, which gives additional information at zero additional cost is the following:

The number (counted with multiplicity) of isolated zeroes on $\pp^n$ of $n$ polynomials in $n+1$ variables (over an algebraically closed field) is at most the product of their degrees. This bound is achieved if the number of zeroes is finite.

The more elaborate, but in a sense more elementary, "affine" version of Bézout's theorem gives more information on the number of zeroes on the affine space, again requiring no new concepts (although it does remove the information about the number of zeroes on the projective space):

The number (counted with multiplicity) of isolated zeroes of $n$ polynomials in $n$ variables (over an algebraically closed field) is at most the product of their degrees. This bound is achieved if and only if the only common zero of the leading forms of these polynomials is the origin.

Similar observations apply to common formulations of other "Bézout-type" theorems, e.g. the Bernstein-Kushnirenko theorem. In fact a formulation of the latter theorem in a recently posted expository article on arxiv is what prompted this question. I might add it as an answer later if no one takes on the mantle.

$\DeclareMathOperator{\pp}{\mathbb{P}}$My aim here is to have a collection of "natural" not-so-common reformulations/extensions of common/well-known results such that

1. the reformulation/extension should provide/facilitate new insights, and
2. it should not require concepts which are more "technical" than the original/common formulation.

For example, a (the most?) common formulation of Bézout's theorem is the following:

The number (counted with multiplicity) of zeroes on $\pp^n$ of $n$ homogeneous polynomials in $n+1$ variables (over an algebraically closed field) is the product of their degrees provided the number of solutions is finite.

Expressed this way, Bézout's theorem says nothing when the number of zeroes is infinite, and on this site there are questions caused by the resulting incomplete understanding. A more natural formulation, which gives additional information at zero additional cost is the following:

The number (counted with multiplicity) of isolated zeroes on $\pp^n$ of $n$ homogeneous polynomials in $n+1$ variables (over an algebraically closed field) is at most the product of their degrees. This bound is achieved if the number of zeroes is finite.

The more elaborate, but in a sense more elementary, "affine" version of Bézout's theorem gives more information on the number of zeroes on the affine space, again requiring no new concepts (although it does remove the information about the number of zeroes on the projective space):

The number (counted with multiplicity) of isolated zeroes of $n$ polynomials in $n$ variables (over an algebraically closed field) is at most the product of their degrees. This bound is achieved if and only if the only common zero of the leading forms of these polynomials is the origin.

Similar observations apply to common formulations of other "Bézout-type" theorems, e.g. the Bernstein-Kushnirenko theorem. In fact a formulation of the latter theorem in a recently posted expository article on arxiv is what prompted this question. I might add it as an answer later if no one takes on the mantle.