MATHEMATICS

E. P. DOLZHENKO

SOME ESTIMATES CONCERNING ALGEBRAIC HYPERSURFACES AND DERIVATIVES OF RATIONAL FUNCTIONS

(Presented by Academician A. N. Kolmogorov on 6 IV 1961)

1, 0. Let \(P_n(x)=P_n(x_1,x_2,\ldots,x_k)\not\equiv 0\) be a real algebraic polynomial of degree \(n\) in the aggregate of variables \(x_1,x_2,\ldots,x_k\) \((k\geqslant 1)\); let \(D_k\) be the closed unit cube of the \(k\)-dimensional real Euclidean space \(E_k\) (the faces of \(D_k\) need not be parallel to the coordinate planes); let \(\Gamma_k(P_n)\) be the hypersurface lying in the space \(E_k\) and defined by the equation \(P_n(x_1,x_2,\ldots,x_k)=0\); let \(\gamma_k(P_n)=\Gamma_k(P_n)\cap D_k\); and let \(N_k(\varepsilon,P_n)\) be the least number of points forming, for a given \(\varepsilon>0\), an \(\varepsilon\)-net on \(\gamma_k(P_n)\).

1,1. Theorem 1. For each \(k=1,2,\ldots\) there exists a constant \(N_k\) \(\bigl(1\leqslant N_k\leqslant 2^{1/2(k+4)(k-1)}\bigr)\), depending only on \(k\), such that
\[ N_k(\varepsilon,P_n)\leqslant N_k n\bigl(\varepsilon^{1-k}+n^{k-1}\bigr) \]
for any polynomial \(P_n\) and any \(\varepsilon>0\).

The theorem can be proved by induction on \(k\), using the fact that \(\Gamma_k(P_n)\) has no more than \(2nk^2\) bounded connected components \(({}^1,{}^2)\).

Corollary 1. The \(k\)-dimensional volume of the \(\varepsilon\)-neighborhood of the set \(\gamma_k(P_n)\) does not exceed \(V_k[\varepsilon n+(\varepsilon n)^k]\), where the constant \(V_k\) depends only on \(k\).

1,2. We introduce new notation. Let \(\Omega_k\) be the \(k\)-dimensional unit ball belonging to \(E_k\), with center at the origin, and also the \(k\)-dimensional volume of this ball:
\[ \Omega_k=\pi^{n/2}/\Gamma(n/2+1); \]
let \(\omega_k\) be the surface of the ball \(\Omega_k\), and also the \((k-1)\)-dimensional area of this surface:
\[ \omega_k=2\pi^{n/2}/\Gamma(n/2); \]
let \(e\) be a point of the sphere \(\omega_k\), and also the unit vector issuing from the origin and having the point \(e\) as its endpoint; let \(\sigma(e)\) be the \((k-1)\)-dimensional hyperplane passing through the origin and having the vector \(e\) as its normal vector; let \(L(T,e)\) be the straight line passing through the point \(T\in\sigma(e)\) orthogonally to \(\sigma(e)\); let \(S\) be a \((k-1)\)-dimensional hypersurface lying in \(E_k\), and also the \((k-1)\)-dimensional measure of this hypersurface; and let \(\nu(T,e,S)\) be the number of connected components of the set \(S\cap L(T,e)\) \((0\leqslant \nu(T,e,S)\leqslant \infty)\).

Lemma. For a piecewise smooth \(*\) \((k-1)\)-dimensional hypersurface \(S\),
\[ S=\frac{1}{2\Omega_{k-1}}\int_{\omega_k} d_e\omega_k \int_{\sigma(e)} \nu(T,e,S)\,d_T\sigma(e) \quad **. \]

The proof of the lemma for the case when \(S\) is a piece of a plane is obtained easily if one observes that in this case
\[ \int_{\sigma(e)} \nu(T,e,S)\,d_T\sigma(e) \]
is the \((k-1)\)-dimensional area of the orthogonal projection of \(S\) onto \(\sigma(e)\). Since the validity of the lemma for polyhedral surfaces already follows from this, to prove it in the general case it remains only to approximate the piecewise smooth hypersurface \(S\) by a polyhedral surface \(\Sigma\) and pass to the limit. This lemma is a generalization of Theorem 22 from \(({}^3)\).

* A \((k-1)\)-dimensional hypersurface is called piecewise smooth if it can be subdivided into a finite number of pieces with a continuously varying \((k-1)\)-dimensional tangent hyperplane.

** The subscript on the differential sign indicates the variable of integration.

Taking into account that every line not lying on \(\Gamma_k(P_n)\) intersects \(\Gamma_k(P_n)\) in no more than \(n\) points, and that every line intersecting some domain \(G\) of the space \(E_k\) intersects the boundary \(S\) of this domain in no fewer than two places, as a consequence of the lemma we obtain the theorem:

Theorem 2. The \((k-1)\)-dimensional area of that part of the surface \(\Gamma_k(P_n)\) which falls in the domain \(G \subset E_k\) does not exceed \(\frac12 nS\), where \(S\) is the \((k-1)\)-dimensional area of the boundary of the domain \(G\).

Corollary 1. The \((k-1)\)-dimensional area of the surface \(\gamma_k(P_n)\) does not exceed \(kn\).

Corollary 2. The \((k-1)\)-dimensional area of that part of the hypersurface \(\Gamma_k(P_n)\) which falls in an ellipsoid with semiaxes \(a_1, a_2, \ldots, a_k\) does not exceed
\[ \pi n^{\,n/2} a_1 a_2 \cdots a_k / \Gamma(n/2). \]
For even \(n\) this estimate is sharp for every set \(k, a_1, a_2, \ldots, a_k\).

The sharpness of the estimate is indicated by the example of the polynomial
\[ P_n(x_1,x_2,\ldots,x_k)=\prod_{j=1}^{n/2}\left[\sum_{i=1}^{k}\left(\frac{x_i}{b_{ji}}\right)^2-1\right], \]
where \(b_{ji}\) are sufficiently close to \(a_i\) \((b_i<a_i)\).

In conclusion of this subsection we note that estimates analogous to those given above can be obtained by the same methods also for algebraic hypersurfaces (lying in \(E_k\)) having dimension less than \(k-1\).

1.3. Let
\[ \widetilde P_n(z)=z^n+a_1z^{n-1}+\cdots+a_n \]
be a polynomial in the complex variable \(z\) with leading coefficient equal to \(1\); \(C\) is a positive constant. With the aid of the lemma given above and the theorems on transfinite diameter \((^4)\), the following is proved:

Theorem 3. The length of the curve \(\Gamma:\ |\widetilde P_n(z)|=C\) does not exceed
\[ 4\pi n\sqrt[n]{C}. \]
The constant \(4\pi\) cannot, for any \(n\) and \(C\), be replaced by \(2\).

2.0. We give estimates of derivatives of rational functions.

It is well known \((^5)\) that the derivative of an algebraic polynomial \(P(x)\) on the interval \([a,b]\) can be estimated in terms of the maximum of \(|P(x)|\) on this interval, the degree of \(P(x)\), and the length of \([a,b]\). From the example of the function \(R_2(x)=\varepsilon/(x^2+\varepsilon)\) it is easy to see that the derivative of a rational fraction \(R(x)\), in general, can no longer be estimated in a similar manner. However, if one “throws out” from the number axis, in a special way, some set \(e\) of prescribed measure \(\delta>0\), then outside this set the derivative of \(R(x)\) can already be estimated in terms of the order of \(R(x)\) and the maximum of its modulus \((^6)\). In this case the number \(\delta\) enters into the estimate.

Below, with the aid of a simple method consisting in estimating the integral of a certain power of the modulus of \(R'(z)\) in terms of the order of \(R(z)\) and the maximum of its modulus, sharp estimates of this kind are obtained for rational functions of a real, as well as of a complex, variable. These estimates are also useful in considering polynomials \(P(z)\) in cases where the known estimates for \(P'(x)\) cease to be applicable (as, for example, in the case when the set \(E\) in the complex plane on which the derivative of the polynomial \(P(z)\) is estimated in terms of its degree and the maximum of its modulus has a sufficiently general nature—say, is a nowhere dense set—or, in the case \(E\subset(-\infty,\infty)\), does not consist of a finite number of intervals).

2.1. Let \(R_n(x)\) denote a rational fraction of order not higher than \(n\) (the order of an irreducible rational fraction is the maximum of the degrees of its numerator and denominator); \(g(M,R_n)\) is the set of all points \(x\in(-\infty,\infty)\) for which \(|R_n(x)|\le M\), where \(M\) is a positive constant.

Theorem 4. For any real rational function \(R_n(x)\) and any positive number \(\delta\) there exists a set
\[ E_1(\delta)=E_1(\delta,M,R_n) \]
such that \(\operatorname{mes} E_1(\delta)<\delta\) and, for \(x\in g(M,R_n)\setminus E_1(\delta)\),

\[ |R'_n(x)| \le \frac{2}{\delta}\, nM. \]
The estimate is sharp for every set of \(n, M, \delta\). The equality sign is in fact attained only in the case of the linear function
\[ R_1(x)=\pm \frac{2}{\delta}Mx+b \quad (b=\mathrm{const}). \]

The proof is obtained from the obvious inequality
\[ \int_{g(M,R_n)} |R'_n(x)|\,dx \le 2Mn. \]

It is of interest to compare the estimate given above for \(R'_n(x)\) with S. N. Bernstein’s estimate for the derivative of a polynomial \(P_n(x)\) on \([-1,1]\):
\[ |P'_n(x)| \le (1-x^2)^{-1/2} nM. \]

A generalization of Theorem 4 is:

Theorem 5. Let, on a set \(E \subset (-\infty,\infty)\), a real rational function \(R_n(x)\) (of order not exceeding \(n\)) not exceed the number \(M\) in absolute value. Then for any \(\delta>0\) there exists a set
\[ E(\delta)=E(\delta,M,R_n) \]
(not depending on the set \(E\)) such that \(\operatorname{mes} E(\delta)<\delta\) and, for \(x\in E\setminus E(\delta)\) and all \(p=0,1,2,\ldots\),
\[ |R_n^{(p)}(x)| \le C_p (n/\delta)^p M, \]
where
\[ C_p=p!\,2^{\frac12 p(p+3)}. \]
For each \(p\) the estimate is sharp in order with respect to the totality of \(n, M\), and \(\delta\).

By sharpness in order here is meant the following: there exist positive constants \(A_p\) \((p=0,1,2,\ldots)\) possessing the following properties: for any natural \(n\) and \(p\) and positive \(M\) and \(\delta\) there will be found a set \(E\subset(-\infty,\infty)\) and a real rational fraction \(R_n(x)\) of order \(n\) such that, whatever set \(E'\) of measure \(<\delta\) we take, there will be found a point \(x\in E\setminus E'\) for which
\[ |R_n^{(p)}(x)|>A_p(n/\delta)^pM. \]
It turns out that one may take
\[ A_p=\frac1{1500}\sqrt{p!}, \]
and take the function \(R_n(x)\) to be one and the same for all \(p\).

2.2. Denote by \(G(M,R_n)\) the set of all points \(z\) of the complex plane \(Z\) for which
\[ |R_n(z)| \le M \quad (M>0) \]
(here \(R_n(z)\) has, generally speaking, complex coefficients).

Theorem 6. For any rational function \(R_n(z)\) of order not exceeding \(n\) and any \(\delta>0\) there exists a set
\[ F_1(\delta)=F_1(\delta,M,R_n) \]
such that \(\operatorname{mes}_2 F_1(\delta)<\delta\) and, for
\[ z\in G(M,R_n)\setminus F_1(\delta), \]
\[ |R'_n(z)| \le \sqrt{\pi/\delta}\,\sqrt n\,M. \]
The estimate is sharp for every set of \(n,M,\delta\). The equality sign is in fact attained only for
\[ R_1(z)=e^{i\alpha}\sqrt{\pi/\delta}\,Mz+b \]
(where \(b=\mathrm{const}\), \(\alpha\) is real).

The proof follows from the obvious equality
\[ \int_{G(M,R_n)} |R'_1(z)|^2\,d\sigma=\pi M^2 n. \]

From the proof it is seen that the last theorem (as, incidentally, Theorem 4 also) generalizes to arbitrary \(n\)-sheeted functions. Namely, the following is true:

Theorem \(6'\). Let in a domain \(G\subset Z\) an analytic function \(f(z)\) be no more than \(n\)-sheeted, and let
\[ |f(z)|\le M \]
for \(z\in E\subset G\). Then for every \(\delta>0\) there exists a set \(F(\delta)\) such that \(\operatorname{mes}_2 F(\delta)<\delta\) and, for
\[ z\in E\setminus F(\delta), \]
\[ |f'(z)|\le \sqrt{\pi/\delta}\,M\sqrt n. \]

A generalization of Theorem 6 is

Theorem 7. Let, on a set \(E\) of the complex plane, a rational function \(R_n(z)\) (of order \(n\)) not exceed the number \(M\) in modulus. Then for any \(\delta>0\) there exists a set
\[ F(\delta)=F(\delta,M,R_n) \]
(not depending on \(E\)) such that \(\operatorname{mes}_2 F(\delta)<\delta\) and, for \(z\in E\setminus F(\delta)\) and all \(p=0,1,2,\ldots\),
\[ |R_n^{(p)}(z)| < \widetilde C_p (n/\delta)^{\frac12 p} M, \]
where
\[ \widetilde C_p=\sqrt{\pi^p p!}\,2^{\frac14 p(p+1)}. \]
The estimate is sharp in order with respect to the totality of \(n,M\), and \(\delta\) for each fixed \(p\).

Accuracy in the sense of order is understood here in the same sense as in Theorem 5.

2,3. Below we denote by: \(D_k^0\) the unit cube of the \(k\)-dimensional real Euclidean space
\[ E_k=\{x:x=(x_1,x_2,\ldots,x_k)\}\quad (k\geqslant 1) \]
with faces parallel to the coordinate hyperplanes; \(Ц_k\) the unit \(k\)-cylinder of the complex \(k\)-dimensional Euclidean space
\[ \widetilde E_k=\{z:z=(z_1,z_2,\ldots,z_k)\}\quad (k\geqslant 1), \]
i.e.
\[ Ц_k=\{z:z=(z_1,z_2,\ldots,z_k);\ |z_i|\leqslant 1,\ i=1,2,\ldots,k\}; \]
\(R_n(y)=R_n(y_1,y_2,\ldots,y_k)\) a rational fraction in the variables \(y_1,y_2,\ldots,y_k\) of order not exceeding \(n\) in each variable. With the aid of Theorems 5 and 7, induction on \(k\) proves Theorems 8 and 9.

Theorem 8. Let, on a set \(E\subset D_k^0\), the real rational function \(R_n(x_1,x_2,\ldots,x_k)\) be, in absolute value, not greater than \(M\). Then for any \(\delta>0\) there exists a set \(E(\delta)=E(\delta,M,R_n)\) such that \(\operatorname{mes}_k E(\delta)<\delta\), and for \((x_1,x_2,\ldots,x_k)\in E\setminus E(\delta)\) and all \(p_i\geqslant 0\) \((i=1,2,\ldots,k)\), \(\sum p_i=p\), the estimate
\[ \left|\frac{\partial^p R_n(x_1,x_2,\ldots,x_k)} {\partial x_1^{p_1}\partial x_2^{p_2}\cdots\partial x_k^{p_k}}\right| \leqslant C_{k,p}(n/\delta)^p M, \]
holds, where
\[ C_{k,p}=p!\,2^{1/2}p(p+3)\,2^{(k-1)p(p+1)}. \]
The estimate is sharp in order jointly in \(n,M,\delta\) (for \(\delta<1/2\)) for every choice of \(k,p_1,p_2,\ldots,p_k\).

Theorem 9. Let, on a set \(\widetilde E\subset Ц_k\), the function \(R_n(z_1,z_2,\ldots,z_k)\) be, in modulus, not greater than the number \(M\). Then for any \(\delta>0\) there exists a set \(F(\delta)=F(\delta,M,R_n)\) such that \(\operatorname{mes}_{2k}F(\delta)<\delta\), and for \((z_1,z_2,\ldots,z_k)\in \widetilde E\setminus F(\delta)\) and all \(p_i\geqslant 0\) \((i=1,2,\ldots,k)\), \(\sum p_i=p\), the inequality
\[ \left|\frac{\partial^p R_n(z_1,z_2,\ldots,z_k)} {\partial z_1^{p_1}\partial z_2^{p_2}\cdots\partial z_k^{p_k}}\right| \leqslant \widetilde C_{k,p}(n/\delta)^{\frac12 pk^2 2^1 4^k p(p+1)}M, \]
is satisfied, where
\[ \widetilde C_{k,p}=\sqrt{p!}\,\pi^{-\frac12 p k^2 2^1 4^k p(p+1)}. \]
The estimate is sharp in the sense of order in \(n,M,\delta\) \((\delta<1/2)\) for every choice of \(k,p_1,p_2,\ldots,p_k\).

2,4. By somewhat modifying A. Cartan’s method for estimating from below the modulus of a polynomial (7) (applied in (6a) in estimating the derivative of a rational function of a real variable), one can prove the following theorem:

Theorem 10. For any rational function \(R_n(z)\) of order \(n\) and any \(\delta>0\) one can find a set \(e(\delta,R_n)\) such that \(\operatorname{mes}_2 e(\delta,R_n)<\delta\), and for \(z\notin e(\delta,R_n)\) and all \(p=0,1,2,\ldots\),
\[ \left|R_n^{(p)}(z)\right|\leqslant S_p(n/\sqrt{\delta})^p |R_n(z)|, \]
where
\[ S_p=p!\,2^{\frac14 p(p+15)}. \]
For each fixed \(p\) the estimate is sharp in the sense of order jointly in \(n,\delta\).

The example of the function
\[ R_n(z)=z^{-n} \]
shows the sharpness of the estimate in the sense of order.

Moscow State University
named after M. V. Lomonosov

Received
15 I 1961

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