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Link to @FrançoisBrunault's comment; mild proofreading
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LSpice
LSpice
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Let $V$ be a vector space and $W$ a subspace.

Given a volume on $V$ and a volume on $V/W$, we can define a volume on $W$. (Here "volume" = "translation-invariant measure" but no real measure theory is being used. We could also define a volume as an element of the top wedge power of the dual vector space.)

Geometrically, this says that given a basis $v_1,\dots, v_m$$v_1,\dotsc, v_m$ of $W$ which extends to a basis $v_{m+1},\dots, v_n$$v_1,\dotsc, v_n$ of $V$, the volume of the parallelepiped spanned by $v_1,\dots, v_m$$v_1,\dotsc, v_m$ in $W$ is equal to the volume of the parallelepiped spanned by $v_1,\dots, v_n$$v_1,\dotsc, v_n$ in $V$ divided bytimes the volume of the parallelepiped spanned by $v_{m+1},\dots,v_n$$v_{m+1},\dotsc,v_n$ in $W/V$$V/W$. (The same thing works with any shape in $V$ whose fibers under the projection map to $W/V$$V/W$ are all translates of a fixed shape in $W$.)

We can express this algebraically with top forms as well, as François Brunault explains in the commentscomments.

For $V$ the vector space tuplesof tuples $x_1,\dots, x_{r+s}$$(x_1,\dotsc, x_{r+s})$, and $W$ the subspace $\sum_i x_i=0$, it is natural to identify $W/V$$V/W$ with $\mathbb R$ under the map $\sum_i x_i$ and take the standard volume on $\mathbb R$. When $x_i$ are the logs of the individual coordinates, this identification corresponds to the norm map.

The reason this is the most commonly used normalization in number theory problem has to do with the fact that the norm map is used often in number theory. In particular, the norm is used in defining the zeta function, explaining why this approach gives nice formulas for the residue of the zeta function.

Let $V$ be a vector space and $W$ a subspace.

Given a volume on $V$ and a volume on $V/W$, we can define a volume on $W$. (Here "volume" = "translation-invariant measure" but no real measure theory is being used. We could also define a volume as an element of the top wedge power of the dual vector space.)

Geometrically, this says that given a basis $v_1,\dots, v_m$ of $W$ which extends to a basis $v_{m+1},\dots, v_n$ of $V$, the volume of the parallelepiped spanned by $v_1,\dots, v_m$ in $W$ is equal to the volume of the parallelepiped spanned by $v_1,\dots, v_n$ in $V$ divided by the volume of the parallelepiped spanned by $v_{m+1},\dots,v_n$ in $W/V$. (The same thing works with any shape in $V$ whose fibers under the projection map to $W/V$ are all translates of a fixed shape in $W$.)

We can express this algebraically with top forms as well, as François Brunault explains in the comments.

For $V$ the vector space tuples $x_1,\dots, x_{r+s}$, and $W$ the subspace $\sum_i x_i=0$, it is natural to identify $W/V$ with $\mathbb R$ under the map $\sum_i x_i$ and take the standard volume on $\mathbb R$. When $x_i$ are the logs of the individual coordinates, this identification corresponds to the norm map.

The reason this is the most commonly used normalization in number theory problem has to do with the fact that the norm map is used often in number theory. In particular, the norm is used in defining the zeta function, explaining why this approach gives nice formulas for the residue of the zeta function.

Let $V$ be a vector space and $W$ a subspace.

Given a volume on $V$ and a volume on $V/W$, we can define a volume on $W$. (Here "volume" = "translation-invariant measure" but no real measure theory is being used. We could also define a volume as an element of the top wedge power of the dual vector space.)

Geometrically, this says that given a basis $v_1,\dotsc, v_m$ of $W$ which extends to a basis $v_1,\dotsc, v_n$ of $V$, the volume of the parallelepiped spanned by $v_1,\dotsc, v_m$ in $W$ is equal to the volume of the parallelepiped spanned by $v_1,\dotsc, v_n$ in $V$ times the volume of the parallelepiped spanned by $v_{m+1},\dotsc,v_n$ in $V/W$. (The same thing works with any shape in $V$ whose fibers under the projection map to $V/W$ are all translates of a fixed shape in $W$.)

We can express this algebraically with top forms as well, as François Brunault explains in the comments.

For $V$ the vector space of tuples $(x_1,\dotsc, x_{r+s})$, and $W$ the subspace $\sum_i x_i=0$, it is natural to identify $V/W$ with $\mathbb R$ under the map $\sum_i x_i$ and take the standard volume on $\mathbb R$. When $x_i$ are the logs of the individual coordinates, this identification corresponds to the norm map.

The reason this is the most commonly used normalization in number theory problem has to do with the fact that the norm map is used often in number theory. In particular, the norm is used in defining the zeta function, explaining why this approach gives nice formulas for the residue of the zeta function.

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Will Sawin
Will Sawin
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Let $V$ be a vector space and $W$ a subspace.

Given a volume on $V$ and a volume on $V/W$, we can define a volume on $W$. (Here "volume" = "translation-invariant measure" but no real measure theory is being used. We could also define a volume as an element of the top wedge power of the dual vector space.)

Geometrically, this says that given a basis $v_1,\dots, v_m$ of $W$ which extends to a basis $v_{m+1},\dots, v_n$ of $V$, the volume of the parallelepiped spanned by $v_1,\dots, v_m$ in $W$ is equal to the volume of the parallelepiped spanned by $v_1,\dots, v_n$ in $V$ divided by the volume of the parallelepiped spanned by $v_{m+1},\dots,v_n$ in $W/V$. (The same thing works with any shape in $V$ whose fibers under the projection map to $W/V$ are all translates of a fixed shape in $W$.)

We can express this algebraically with top forms as well, as François Brunault explains in the comments.

For $V$ the vector space tuples $x_1,\dots, x_{r+s}$, and $W$ the subspace $\sum_i x_i=0$, it is natural to identify $W/V$ with $\mathbb R$ under the map $\sum_i x_i$ and take the standard volume on $\mathbb R$. When $x_i$ are the logs of the individual coordinates, this identification corresponds to the norm map.

The reason this is the most commonly used normalization in number theory problem has to do with the fact that the norm map is used often in number theory. In particular, the norm is used in defining the zeta function, explaining why this approach gives nice formulas for the residue of the zeta function.