Return to Answer

For any $n$, there is an $n$-ary smooth function that is not a composition of smooth functions of lower arity; according to this answer to a very similar question, this is due to Vitushkin (at least for $n=3$). Here is a simple but rather inexplicit proof (adapted from this paper of Akashi and Kodama). Suppose that a smooth n-ary function $f$ can be expressed as a composition of smooth functions $g_i$ of lower arity (WLOG, all of arity $n-1$). Then for each $k$, the $k$-jet of $f$ at any point is determined by the $k$-jets of the $g_i$ at corresponding points. But dimension of the space of $n$-ary $k$-jets grows like $k^n$, while the dimension of the space of $(n-1)$-ary $k$-jets grows like $k^{n-1}$. It follows that for any particular way to compose lower arity functions to get an $n$-ary function, most $k$-jets of $n$-ary functions cannot be so obtained for $k$ sufficiently large. Since there are only countably many such ways to compose, we can use bump functions to construct a single $f$ that has jets at different points that fail for all of them. Actually, by a diagonal argument we can find a single $n$-ary $\infty$-jet (i.e., power series) that is not a composition of jets of lower arity, and every $\infty$-jet can be realized by a smooth function.

For any $n$, there is an $n$-ary smooth function that is not a composition of smooth functions of lower arity; according to this answer to a very similar question, this is due to Vitushkin (at least for $n=3$). Here is a simple but rather inexplicit proof (adapted from this paper of Akashi and Kodama). Suppose that a smooth n-ary function $f$ can be expressed as a composition of smooth functions $g_i$ of lower arity (WLOG, all of arity $n-1$). Then for each $k$, the $k$-jet of $f$ at any point is determined by the $k$-jets of the $g_i$ at corresponding points. But dimension of the space of $n$-ary $k$-jets grows like $k^n$, while the dimension of the space of $(n-1)$-ary $k$-jets grows like $k^{n-1}$. It follows that for any particular way to compose lower arity functions to get an $n$-ary function, most $k$-jets of $n$-ary functions cannot be so obtained for $k$ sufficiently large. Since there are only countably many such ways to compose, we can use bump functions to construct a single $f$ that has jets at different points that fail for all of them. Actually, by a diagonal argument we can find a single $n$-ary $\infty$-jet (i.e., power series) that is not a composition of jets of lower arity, and every $\infty$-jet can be realized by a smooth function.

For any $n$, there is an $n$-ary smooth function that is not a composition of smooth functions of lower arity; according to this answer to a very similar question, this is due to Vitushkin (at least for $n=3$). Here is a simple but rather inexplicit proof (adapted from this paper of Akashi and Kodama). Suppose that a smooth n-ary function $f$ can be expressed as a composition of smooth functions $g_i$ of lower arity (WLOG, all of arity $n-1$). Then for each $k$, the $k$-jet of $f$ at any point is determined by the $k$-jets of the $g_i$ at corresponding points. But dimension of the space of $n$-ary $k$-jets grows like $k^n$, while the dimension of the space of $(n-1)$-ary $k$-jets grows like $k^{n-1}$. It follows that for any particular way to compose lower arity functions to get an $n$-ary function, most $k$-jets of $n$-ary functions cannot be so obtained for $k$ sufficiently large. Since there are only countably many such ways to compose, we can use bump functions to construct a single $f$ that has jets at different points that fail for all of them.

For any $n$, there is an $n$-ary smooth function that is not a composition of smooth functions of lower arity; according to this answer to a very similar question, this is due to Vitushkin (at least for $n=3$). Here is a simple but rather inexplicit proof (adapted from this paper of Akashi and Kodama). Suppose that a smooth n-ary function $f$ can be expressed as a composition of smooth functions $g_i$ of lower arity (WLOG, all of arity $n-1$). Then for each $k$, the $k$-jet of $f$ at any point is determined by the $k$-jets of the $g_i$ at corresponding points. But dimension of the space of $n$-ary $k$-jets grows like $k^n$, while the dimension of the space of $(n-1)$-ary $k$-jets grows like $k^{n-1}$. It follows that for any particular way to compose lower arity functions to get an $n$-ary function, most $k$-jets of $n$-ary functions cannot be so obtained for $k$ sufficiently large. Since there are only countably many such ways to compose, we can use bump functions to construct a single $f$ that has jets at different points that fail for all of them. Actually, by a diagonal argument we can find a single $n$-ary $\infty$-jet (i.e., power series) that is not a composition of jets of lower arity, and every $\infty$-jet can be realized by a smooth function.

Not every 3-ary smooth function is a composition of 2-ary smooth functions; according to this answer to a very similar question, this is due to Vitushkin. Here is a very simple but nonconstructive argument for it (taken from this paper of Akashi and Kodama) which easily generalizes to show that there are smooth functions of any arity which are not a composition of smooth functions of lower arity. Suppose that a smooth 3-ary function $f$ can be expressed in terms of smooth 2-ary functions $g_i$. Then for each $n$, the $n$-jet of $f$ at any point is determined by the $n$-jets of the $g_i$ at corresponding points. But dimension of the space of 3-ary $n$-jets grows like $n^3$, while the dimension of the space of 2-ary $n$-jets grows like $n^2$. It follows that for any particular way to compose 2-ary functions to get a 3-ary function, most $n$-jets of 3-ary functions cannot be so obtained for $n$ sufficiently large. Since there are only countably many such ways to compose, we can use bump functions to construct a single $f$ that has jets at different points that fail for all of them.

For any $n$, there is an $n$-ary smooth function that is not a composition of smooth functions of lower arity; according to this answer to a very similar question, this is due to Vitushkin (at least for $n=3$). Here is a simple but rather inexplicit proof (adapted from this paper of Akashi and Kodama). Suppose that a smooth n-ary function $f$ can be expressed as a composition of smooth functions $g_i$ of lower arity (WLOG, all of arity $n-1$). Then for each $k$, the $k$-jet of $f$ at any point is determined by the $k$-jets of the $g_i$ at corresponding points. But dimension of the space of $n$-ary $k$-jets grows like $k^n$, while the dimension of the space of $(n-1)$-ary $k$-jets grows like $k^{n-1}$. It follows that for any particular way to compose lower arity functions to get an $n$-ary function, most $k$-jets of $n$-ary functions cannot be so obtained for $k$ sufficiently large. Since there are only countably many such ways to compose, we can use bump functions to construct a single $f$ that has jets at different points that fail for all of them.