Let \(A[\mathbf{x}] = A[x_{1}, \cdots, x_{n}\)] be the polynomial ring in n variables over an integral domain \(A\), \(D\) an \(A\)-derivation of \(A[\mathbf{x}]\) and denote \[L^{D}_{ij} := D(x_{i}) x_{j} - D(x_{j}) x_{i}, \quad \text{ for each } i, j \in \{1, \cdots, n\}.\]

Theorem 1 [Nowicki 1, Conjecture 6.9.10, Kuroda's Proof]

Assume \(k[\mathbf y]=k[y_1,\ldots,y_n]\) is a polynomial ring in \(n\) variables over a field \(k\) of characteristic zero. If \(\Delta_{n}\) is the \(k[\mathbf{y}]\)-derivation of \(k[\mathbf{y}][\mathbf{x}]\) defined by \(\Delta_{n}(x_{i}) = y_{i}\) for \(i = 1, \cdots, n\), then

\[ \ker\Delta_n= k[\mathbf y]\big[L^{\Delta_n}_{ij}\big] = k[\mathbf y]\big[y_i x_j-y_jx_i:1\le i < j \le n\big]. \]

Equivalently, the ring of constants of \(\Delta_n\) is generated by the \(2\times 2\)-minors \(y_i x_j-y_jx_i,1\le i < j \le n\) over \(k[\mathbf y]\).

Proof

Induction step

We prove the conjecture by induction on \(n\). The assertion is clear when \(n=1\). Assume that \(n\geq2\), and for each \(l\leq n\), let \(S_l\) be the set of \(L_{i,j}:=L_{i,j}^{\Delta_n}\) with \(1\leq i<j\leq l\). By the induction hypothesis, \(\ker\Delta_{n-1}\) is generated by \(S_{n-1}\) over \(k[\mathbf y']:=k[y_1,\ldots,y_{n-1}]\), since \(L_{i,j}^{\Delta_{n-1}}=L_{i,j}^{\Delta_n}\) for each \(i,j\). The \(k[\mathbf y']\)-derivation \(\Delta_{n-1}\) naturally extends to a \(k[\mathbf y]\)-derivation \((\Delta_{n-1})_{k[\mathbf y]}\) of \(k[\mathbf y][\mathbf x']:=k[\mathbf y][x_1,\ldots,x_{n-1}]\). Then \((\Delta_{n-1})_{k[\mathbf y]}=\Delta_n|_{k[\mathbf y][\mathbf x']}\), so \(\ker(\Delta_{n-1})_{k[\mathbf y]}=k[\mathbf y][\mathbf x']\cap\ker\Delta_n\). Moreover, \(\ker(\Delta_{n-1})_{k[\mathbf y]}=k[\mathbf y]\otimes_{k[\mathbf y']}\ker\Delta_{n-1}\), since \(k[\mathbf y]\) is flat over \(k[\mathbf y']\). Thus,

\[ k[\mathbf y][\mathbf x']\cap\ker\Delta_n=k[\mathbf y][S_{n-1}]. \tag{1} \]

The \(\Gamma\)-grading

Let \(\mathbf e_1,\ldots,\mathbf e_n\) be the coordinate unit vectors of \(\mathbb Z^n\), let \(M\) be the \(\mathbb Z\)-submodule of \((\mathbb Z^n)^2\) generated by \((\mathbf e_j-\mathbf e_i,\mathbf e_i-\mathbf e_j)\) for \(1\leq i<j\leq n\), and set \(\Gamma=(\mathbb Z^n)^2/M\). We define \(\Gamma\)-gradings on \(k[\mathbf y][\mathbf x]\) and \(k[\mathbf y^{\pm1}][\mathbf x]:=k[\mathbf y][\mathbf x][(y_1\cdots y_n)^{-1}]\) as follows. Recall that a \(k\)-algebra \(R\) is \(\Gamma\)-graded if there are \(k\)-vector subspaces \(R_\gamma\), \(\gamma\in\Gamma\), such that \(R=\bigoplus_{\gamma\in\Gamma}R_\gamma\) and \(R_\gamma R_\mu\subset R_{\gamma+\mu}\) for all \(\gamma,\mu\in\Gamma\). Let \(\mathbb Z_{\geq0}\) denote the nonnegative integers, and write \(\mathbf y^a=y_1^{a_1}\cdots y_n^{a_n}\) and \(\mathbf x^b=x_1^{b_1}\cdots x_n^{b_n}\) for \(a=(a_1,\ldots,a_n)\) and \(b=(b_1,\ldots,b_n)\). For each \(\gamma\in\Gamma\), define \(k[\mathbf y][\mathbf x]_\gamma\) (resp. \(k[\mathbf y^{\pm1}][\mathbf x]_\gamma\)) to be the \(k\)-vector space generated by \(\mathbf y^a\mathbf x^b\) with \(a,b\in(\mathbb Z_{\geq0})^n\) (resp. \(a\in\mathbb Z^n\) and \(b\in(\mathbb Z_{\geq0})^n\)) whose image in \(\Gamma\) is \(\gamma\). Note that \(\Delta_n(k[\mathbf y][\mathbf x]_\gamma)\subset k[\mathbf y][\mathbf x]_{\gamma-\delta}\) for each \(\gamma\in\Gamma\), where \(\delta\) is the image of \((-\mathbf e_n,\mathbf e_n)\) in \(\Gamma\). Hence \(\ker\Delta_n=\bigoplus_{\gamma\in\Gamma}(k[\mathbf y][\mathbf x]_\gamma\cap\ker\Delta_n)\).

Reduction to a homogeneous element

Thus, it is enough to show that every \(0\neq\Phi\in k[\mathbf y][\mathbf x]_\gamma\cap\ker\Delta_n\) belongs to \(k[\mathbf y][S_n]\). Choose \(a=(a_1,\ldots,a_n)\in\mathbb Z^n\) and \(l\in\mathbb Z_{\geq0}\) such that the image of \((a,l\mathbf e_n)\) in \(\Gamma\) is \(\gamma\). Let \(m\) be the \(x_n\)-degree of \(\Phi\), with \(0\leq m\leq l\), and let \(\phi\in k[\mathbf y][\mathbf x']\) be the coefficient of \(x_n^m\) in \(\Phi\). Then \(\phi\in k[\mathbf y][\mathbf x]_\mu\), where \(\mu\) is the image of \((a,(l-m)\mathbf e_n)\) in \(\Gamma\). Furthermore, \(0=\Delta_n(\Phi)=\Delta_n(\phi)x_n^m+m\phi y_nx_n^{m-1}+\Delta_n(\Phi-\phi x_n^m)\). Since the last two terms have \(x_n\)-degree at most \(m-1\), we have \(\Delta_n(\phi)=0\). Hence \(\phi\in k[\mathbf y][S_{n-1}]\) by (1).

Write \(\phi=\sum_{b,\mathbf u}r'_{b,\mathbf u}\mathbf y^b\widehat{\mathbf y}^{-\mathbf u}L^{\mathbf u}\), where \(b\in(\mathbb Z_{\geq0})^n\), \(\mathbf u=(u_{i,j})_{1\leq i<j\leq n-1}\) with \(u_{i,j}\in\mathbb Z_{\geq0}\), and \(r'_{b,\mathbf u}\in k\). Here \(\widehat{\mathbf y}^{-\mathbf u}:=\prod_{1\leq i<j\leq n-1}(y_iy_j)^{-u_{i,j}}\) and \(L^{\mathbf u}:=\prod_{1\leq i<j\leq n-1}L_{i,j}^{u_{i,j}}\). We may assume that \(r'_{b,\mathbf u}=0\) whenever \(\mathbf y^b\widehat{\mathbf y}^{-\mathbf u}\notin k[\mathbf y]\). Let \(\eta(b,\mathbf u)\) be the image of \((b-|\mathbf u|\mathbf e_n,|\mathbf u|\mathbf e_n)\) in \(\Gamma\), where \(|\mathbf u|:=\sum_{1\leq i<j\leq n-1}u_{i,j}\). Then \(\mathbf y^b\widehat{\mathbf y}^{-\mathbf u}L^{\mathbf u}\in k[\mathbf y^{\pm1}][\mathbf x]_{\eta(b,\mathbf u)}\), since \((y_iy_j)^{-1}L_{i,j}\in k[\mathbf y^{\pm1}][\mathbf x]_\delta\) for each \(i,j\).

Since \(\phi\in k[\mathbf y][\mathbf x]_\mu\) and \(\mu\) is the image of \((a,(l-m)\mathbf e_n)\), we may assume that \(r'_{b,\mathbf u}=0\) unless \(|\mathbf u|=l-m\) and \(b=a+(l-m)\mathbf e_n\). For each \(\mathbf u\) with \(r_{\mathbf u}:=r'_{a+(l-m)\mathbf e_n,\mathbf u}\neq0\), write \(\mathbf y^ay_n^{l-m}\widehat{\mathbf y}^{-\mathbf u}=y_1^{\rho_1(\mathbf u)}\cdots y_{n-1}^{\rho_{n-1}(\mathbf u)}y_n^s\), where \(\rho_i(\mathbf u)\in\mathbb Z_{\geq0}\) for \(i=1,\ldots,n-1\) and \(s=a_n+l-m\). Then \(\phi=y_n^s\sum_{\mathbf u}r_{\mathbf u}y_1^{\rho_1(\mathbf u)}\cdots y_{n-1}^{\rho_{n-1}(\mathbf u)}L^{\mathbf u}\). Since \(|\mathbf u|=l-m\), it follows that

\[ \sum_{i=1}^{n-1}\rho_i(\mathbf u)=\sum_{i=1}^{n-1}a_i-2(l-m). \tag{2} \]

The minimal-degree argument

Now we show that \(\Phi\in k[\mathbf y][S_n]\) by contradiction. Replacing \(\Phi\) if necessary, we may assume that \(m\) is minimal among the \(x_n\)-degrees of elements of \(\ker\Delta_n\setminus k[\mathbf y][S_n]\). To obtain a contradiction, it suffices to prove

\[ m\geq2l-\sum_{i=1}^{n-1}a_i. \tag{3} \]

Indeed, (3) and (2) imply \(\sum_{i=1}^{n-1}\rho_i(\mathbf u)\geq m\). Hence, for each \(\mathbf u\), there exist integers \(0\leq\rho_i'(\mathbf u)\leq\rho_i(\mathbf u)\) such that \(\sum_{i=1}^{n-1}\rho_i'(\mathbf u)=m\). Define

\[ \Phi':=y_n^s\sum_{\mathbf u}r_{\mathbf u}L^{\mathbf u}\prod_{i=1}^{n-1}y_i^{\rho_i(\mathbf u)-\rho_i'(\mathbf u)}L_{i,n}^{\rho_i'(\mathbf u)} =y_n^s\sum_{\mathbf u}r_{\mathbf u}L^{\mathbf u}\prod_{i=1}^{n-1}y_i^{\rho_i(\mathbf u)-\rho_i'(\mathbf u)}(y_ix_n-y_nx_i)^{\rho_i'(\mathbf u)}. \]

Then \(\Phi'\in k[\mathbf y][S_n]\) has \(x_n\)-degree \(m\), and the coefficient of \(x_n^m\) is \(\phi\). Therefore, the \(x_n\)-degree of \(\Phi-\Phi'\) is less than \(m\). Since \(\Phi-\Phi'\in\ker\Delta_n\setminus k[\mathbf y][S_n]\), this contradicts the minimality of \(m\).

Proof of the degree bound

We first record a standard fact about locally nilpotent derivations.

Lemma (Factorial closure)

Let \(D\) be a locally nilpotent derivation of an integral domain \(R\) of characteristic zero. Then \(\ker D\) is factorially closed in \(R\): if \(0\neq fg\in\ker D\), then \(f,g\in\ker D\).

Proof

For \(0\neq h\in R\), define \(\deg_D(h):=\max\{r\geq0:D^r(h)\neq0\}\), which is finite because \(D\) is locally nilpotent. If \(a=\deg_D(f)\) and \(b=\deg_D(g)\), then the Leibniz formula gives \(D^{a+b}(fg)=\binom{a+b}{a}D^a(f)D^b(g)\neq0\), while \(D^{a+b+1}(fg)=0\). Hence \(\deg_D(fg)=a+b\). If \(fg\in\ker D\), then \(\deg_D(fg)=0\), so \(a=b=0\), and therefore \(f,g\in\ker D\). \(\square\)

It remains to establish (3) for every nonzero homogeneous element \(\Phi\in\ker\Delta_n\). Suppose, to the contrary, that (3) fails, and choose such a \(\Phi\) with \(m\) minimal. Then \(t:=2l-\sum_{i=1}^{n-1}a_i-m>0\), and by (2), \(\sum_{i=1}^{n-1}\rho_i(\mathbf u)=m-t\) for each \(\mathbf u\). Hence, the \(x_n\)-degree of

\[ \Phi_1:=\sum_{\mathbf u}r_{\mathbf u}L^{\mathbf u}\prod_{i=1}^{n-1}L_{i,n}^{\rho_i(\mathbf u)} =\sum_{\mathbf u}r_{\mathbf u}L^{\mathbf u}\prod_{i=1}^{n-1}(y_ix_n-y_nx_i)^{\rho_i(\mathbf u)} \]

is \(m-t\). The coefficient of \(x_n^{m-t}\) in \(y_n^s\Phi_1\) is \(\phi\), so the coefficient of \(x_n^m\) in \(y_n^s\Phi_1L_{1,n}^t\) is equal to that in \(y_1^t\Phi\). Consequently, the \(x_n\)-degree \(m'\) of \(\Phi_2:=y_1^t\Phi-y_n^s\Phi_1L_{1,n}^t\) is less than \(m\). We claim that \(\Phi_2=0\). Indeed, if \(\gamma'\) is the image of \((a+t\mathbf e_1,l\mathbf e_n)\) in \(\Gamma\) and \((a_1',\ldots,a_n'):=a+t\mathbf e_1\), then \(\Phi_2\in k[\mathbf y][\mathbf x]_{\gamma'}\cap\ker\Delta_n\), and \(2l-\sum_{i=1}^{n-1}a_i'=2l-\sum_{i=1}^{n-1}a_i-t=m>m'\). Thus \(\Phi_2=0\) by the minimality of \(m\).

Hence \(y_1^t\Phi=y_n^s\Phi_1L_{1,n}^t\). Since neither \(y_n\) nor \(L_{1,n}\) is divisible by \(y_1\), it follows that \(\Phi_1\) is divisible by \(y_1\). By the lemma, \(\ker\Delta_n\) is factorially closed. Since \(\Delta_n\) is locally nilpotent, \(\Delta_n(\Phi_1)=0\), \(\Phi_1\neq0\), and \(\Delta_n(x_n)\neq0\), the polynomial \(\Phi_1\) is not divisible by \(x_n\). Setting \(x_n=0\) therefore gives the nonzero polynomial

\[ \sum_{\mathbf u}r_{\mathbf u}L^{\mathbf u}\prod_{i=1}^{n-1}(-y_nx_i)^{\rho_i(\mathbf u)}=(-y_n)^{m-t}\Psi, \qquad \Psi:=\sum_{\mathbf u}r_{\mathbf u}L^{\mathbf u}\prod_{i=1}^{n-1}x_i^{\rho_i(\mathbf u)}. \]

Thus \(\Psi\neq0\), and \(\Psi\) is divisible by \(y_1\), since \(\Phi_1\) is. Define \(\sigma\in\operatorname{Aut}_k k[\mathbf y][\mathbf x]\) by \(\sigma(x_i)=y_i\) and \(\sigma(y_i)=x_i\) for \(i=1,\ldots,n\). Then \(\sigma(\Psi)\) is divisible by \(x_1\). On the other hand, \(\sigma(L_{i,j})=L_{j,i}\) and \(\sigma(x_i)=y_i\) belong to \(\ker\Delta_n\) for each \(i,j\), so \(\sigma(\Psi)\in\ker\Delta_n\). Since \(\ker\Delta_n\) is factorially closed and \(x_1\notin\ker\Delta_n\), this is impossible for the nonzero polynomial \(\sigma(\Psi)\). Therefore (3) holds. Hence \(\Phi\in k[\mathbf y][S_n]\), completing the proof of the conjecture.

A differential approach \({\rm DCF}_0\)

There is a useful way to read Nowicki's theorem inside differential algebra. Let \((\mathcal U,\partial)\models {\rm DCF}_0\), with constant field \(C=\ker\partial\), and consider a tuple satisfying \(\partial y_i=0\) and \(\partial x_i=y_i\) for \(i=1,\ldots,n\). The derivation induced on the polynomial ring \(k[\mathbf y][\mathbf x]\) is exactly Kuroda's derivation \(\Delta_n\). Thus \(\ker\Delta_n\) is the ring of polynomial first integrals of the differential system \(\mathbf y'=0\), \(\mathbf x'=\mathbf y\).

The basic generators appear immediately from this viewpoint. Since \(\partial y_i=0\) and \(\partial x_i=y_i\), we have \(\partial L_{ij}=\partial(y_ix_j-y_jx_i)=y_iy_j-y_jy_i=0\). Hence every \(L_{ij}\) is a differential constant. Geometrically, the system carries the definable action of the additive constant group \((C,+)\) given by \(c\cdot(\mathbf x,\mathbf y)=(\mathbf x+c\mathbf y,\mathbf y)\). The quantities \(y_i\) and \(L_{ij}\) are invariant under this action, and for \(\mathbf y\neq0\) they separate its orbits. In this sense, the minors \(L_{ij}\) provide concrete algebraic coordinates for the quotient by the one-dimensional constant flow.

The rational version is particularly transparent. On the chart \(y_1\neq0\), put \(t=x_1/y_1\). Then \(\partial t=1\), while \(L_{1j}=y_1x_j-y_jx_1\) gives \(x_j=y_jt+L_{1j}/y_1\). Thus the whole differential field is obtained from its constants by adjoining a single element \(t\) with \(t'=1\). Consequently, at the level of rational functions, the first integrals are generated by the \(y_i\) and the \(L_{ij}\):

\[ \operatorname{Frac}(k[\mathbf y][\mathbf x])^{\Delta_n}=k(\mathbf y,L_{ij}). \]

From this perspective, the real content of Nowicki's theorem is not the rational description but the passage back to the polynomial ring. It asserts that no hidden denominators are needed:

\[ k[\mathbf y][\mathbf x]\cap k(\mathbf y,L_{ij})=k[\mathbf y][L_{ij}]. \]

Equivalently, the polynomial first integrals are already generated by the obvious differential constants. This is precisely where Kuroda's argument goes beyond the elementary \(DCF_0\) picture: the differential-field calculation identifies the rational quotient almost immediately, whereas the \(\Gamma\)-grading, the minimal-degree argument, and factorial closure are what eliminate denominators and recover the coordinate ring itself.

There is also a model-theoretic way to phrase the same geometry. For fixed \(\mathbf y\in C^n\), consider the definable equivalence relation \(\mathbf x\sim\mathbf z\) if \(\mathbf z=\mathbf x+c\mathbf y\) for some \(c\in C\). On the locus \(\mathbf y\neq0\), the tuple \((\mathbf y,(L_{ij})_{i<j})\) gives a concrete code for the corresponding orbit. Thus Nowicki's generators may be viewed as polynomial representatives of the natural quotient invariants of a definable \((C,+)\)-action. The theorem says that, in this special linear situation, the model-theoretically natural invariants and the algebraically generated polynomial constants coincide exactly.

References

  1. Andrzej Nowicki, Polynomial Derivations and Their Rings of Constants, Uniwersytet Mikołaja Kopernika, Toruń, 1994.
  2. Shigeru Kuroda, "A Simple Proof of Nowicki's Conjecture on the Kernel of an Elementary Derivation," Tokyo Journal of Mathematics 32 (2009), no. 1, 247-251.