Sergei Yakovenko's blog: on Math and Teaching

Noetherian chains

Computation of the (local intersectional) degree of a phase curve of a polynomial vector field, produced in Lecture 11, is based on the length of the ascending chain of polynomial ideals generated by consecutive derivations.

Let $D:\mathbb C[x_1,\dots,x_n]\to\mathbb C[x_1,\dots,x_n]$ be the Lie derivation of the algebra of polynomials along the vector field $v$. It increases the degrees by at most $d-1$. Let $p_0\in\mathbb C[x]$ be a seed polynomial of degree $\delta\in\mathbb N$ and consider the ascending chain of ideals

$I_0\subseteq I_1\subseteq I_2\subseteq\cdots\subseteq I_k\subseteq\cdots \subseteq\mathbb C[x],\qquad I_k=\left,$

where $p_k=Dp_{k-1},\ k=1,2,\dots$.  By Noetherianity, this chain must eventually stabilize at some step: $I_N=I_{N+1}=\cdots$. In addition to this chain of ideals, one can consider the associated descending chain of algebraic varieties

$\mathbb C^n\supseteq X_0\supseteq X_1\supseteq\cdots\supseteq X_k\supseteq\cdots, \qquad X_k=\{p_0(x)=\cdots=p_k(x)=0\}.$

This chain also stabilizes  no later than on the $N$th step, but may stabilize earlier.  The following properties of these chains can be verified by elementary arguments.

1. The chain of ideals is strictly ascending: If $I_N=I_{N+1}$, then all subsequent ideals in the chain coincide.
2. The chain of varieties may be nonstrictly ascending: e.g., $n=1,\ p_0(x)=x^m,\ D=\frac{\mathrm d}{\mathrm dx}$.
3. The length of the descending chain measures the maximal nontrivial order of contact between the trajectories of $v$ and the hypersurface $X_0=\{p_0=0\}$.

In general, the length of a strictly ascending chain of polynomial ideals generated by the sequence of polynomials of degrees not exceeding an explicit (growing) function of $k$, can be bounded by an algorithmically computable function. However, even in the simplest case where $\deg p_k\le \delta+k(d-1)$ (as above), this function turns out to be the Ackermann generalized exponential, a recursive but not primitively recursive function of $n,d,\delta\in\mathbb N$ which grows faster than any elementary (or primitive recursive) function. It is the algebraic origin of the sequence of polynomials, which allows to establish better results.

Example. Assume that $A:\mathbb C[x]\to\mathbb C[x]$ is an endomorphism of the ring of the polynomials, and instead of the iterations $p_k=Dp_{k-1}$ of the Lie derivation, we consider the sequence $p_k=Ap_{k-1}$. Then analogous chains can be constructed, yet their properties will be slightly different (in a sense, better). In particular, the chain of varieties becomes strictly descending and its length can be relatively simply bounded by simple function of $n,d,\delta$. If the growth rate of $\deg p_k$ is linear (as above), the bound will be double exponential in $n$. However,  in general the growth rate of iterates $A^k p_0$ is exponential, which leads to the bound given by a tower function (iterated exponent) of height $n=\dim x$.

The easiest way to estimate the length of varieties generated by consecutive derivations is based on the explicit Nullstellensatz. By this  theorem, for any polynomial $q\in\mathbb C[x]$ which vanishes on the variety $X\subseteq\mathbb C^n$ which is the zero locus of an ideal $I\subseteq\mathbb C[x]$ there exist a finite power $\rho$ such that $q^\rho\in X$. The number $\rho$ can be explicitly bounded from above (J. Kollar, 1988): if $I$ is generated by polynomials of degree no greater than $m$, then $\rho\leqslant m^n$. Having this bound, for each irreducible component of the variety $X_k$ which does not belong to the stable limit, one can predict, how many steps in can survive before being eliminated.  The resulting upper bound will be double exponential in $n$.

However, a better, more realistic and simple exponential in $n$ upper bound can be achieved by completely different argument.

Example. Assume that $n=2$ and we look at an isolated contact between a (nonsingular) trajectory of a vector field $v$ and an algebraic curve $X_0=X$ of degree $\delta$ at a point $a\in\mathbb C^2$. Consider the local analytic chart in which $v$ is parallel to the $y$-axis and the point $a$ is at the origin. If the curve $X$ has tangency of order $\mu$ with the vertical axis, then its projection on the $x$-axis is locally a ramified covering of order $\mu$. Consider a small bidisk neighborhood of the origin and apply a small analytic perturbation to $X$. The multiple tangency point will be scattered into several points of simple (quadratic) tangency, while the topological covering property will persist. Denote by $\nu$ the number of obtained simple tangencies: at each tangency exactly two leaves of the covering “collide”. Thus the total number of leaves $\mu$ cannot be greater than $2\nu$. The problem thus becomes to estimate $\nu$. However, the set of points of quadratic contact is algebraic: it is defined by the equations $X_1=\{p_0=0, p_1=0\}$ of degrees $\delta$ and $\delta+d-1$, so by the Bezout theorem the number of points does not exceed the product of these two numbers.

To generalize this argument for the multidimensional settings, one has to modify the topological part of the argument dealing with “exactly two leaves of the covering collide”. Instead of just one set $X_1$, one has to consider the sets $X_1,\dots, X_{n-1}$ (recall that $n$ is the dimension of the ambient space), and instead of counting points, one should consider their Euler characteristics. The corresponding combinatorics can be elegantly expressed by the “integration over the Euler characteristic” discovered by O. Viro (1988), while the bounds for the Euler characteristic of algebraic varieties can be bounded by virtue of the J. Milnor’s result (1964).

The result, due to A. Gabrielov and A. Khovanskii, is simple exponential (in $n$) bound, was achieved in 1998. However, for some problems in the analytic number theory (algebraic independence of transcendental numbers) it is important to have a more precise estimate of the maximal tangency order for $d, n$ fixed, but $\delta$ variable and growing to infinity. The most recent achievements in this direction are due to G. Binyamini [4], see below.

Besides, a different (and considerably more difficult) problem arises in the singular context, when one tries to estimate the order of contact of an algebraic hypersurface with a separatrix of a polynomial vector field, an invariant analytic curve (usually non-smooth) which contains a singular point of the vector field $v$.  Here again the most recent breakthroughs are due to Binyamini [5].

References (in addition to those mentioned earlier).

1. A. Gabrielov, A. KhovanksiiMultiplicity of a Noetherian intersection.  Geometry of differential equations, 119–130,
Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, RI, 1998.
2. O. ViroSome integral calculus based on Euler characteristic. Topology and geometry—Rohlin Seminar, 127–138,
Lecture Notes in Math., 1346, Springer, Berlin, 1988.
3. J. Milnor, On the Betti numbers of real varieties.  Proc. Amer. Math. Soc. 15 1964 275–280.
4. G. Binyamini, Multiplicity Estimates: a Morse-theoretic approach, arXiv:1406.1858 (2014).
5. G. Binyamini, Multiplicity estimates, analytic cycles and Newton polytopes, arXiv:1407.1183