## Bernstein’s inequality on algebraic curves.(English)Zbl 0842.26013

In this paper, authors estimate the growth of polynomials on a smooth algebraic curve $$\Gamma= \{(x, y)\in \mathbb{R}^2: y= \psi(x),|x|\leq 1\}$$, where $$Q(x, \psi(x))\equiv 0$$, $$Q$$ being a polynomial with real coefficients. Under the assumptions that (I) $$|\psi(x)|\leq 1$$ for $$|x|\leq 1$$, (II) $$Q$$ has degree at most $$D$$, (III) $$|Q(x, y)|\leq C$$ for $$|x|,|y|\leq 1$$ and (IV) $$|{\partial Q\over \partial y} (x, y)|\geq c> 0$$ for $$(x, y)\in \Gamma$$, they show that, given a polynomial $$P$$ of degree $$d$$ and $$f(x)= P(x, \psi(x))$$, there exists a constant $$C_*> 0$$ depending only on $$d$$, $$D$$, $$C$$, $$c$$ such that $(\text{i})\quad \max_{|x|\leq 1} |f(x)|\leq C_*\max_{|x|\leq {1\over 2}}|f(x)|,\qquad (\text{ii})\quad \max_{|x|\leq 1}|f'(x)|\leq C_* \max_{|x|\leq 1} |f(x)|,$
$\text{(iii)}\qquad \max_{|x|\leq 1} |f(x)|\leq C_* \int^1_{- 1} |f(x)|dx.$ Here, the point is that $$C_*$$ does not depend on $$\Gamma$$. The authors’ main tool is an extension theorem that permits to find, to every polynomial $$P$$ of degree $$d$$, a rational function $$F/G$$ of degree $$d_*$$ such that $$P= F/G$$ on $$\Gamma$$, $$c_*< G< C_*$$ on $$\{|x|, |y|\leq 1\}$$ and $$\max_{|x|, |y|\leq 1} |F|\leq C_*\max_\Gamma |P|$$. One should add that such an extension with $$F/G$$ replaced by a polynomial is, in general, impossible which is shown by the authors by an elementary counterexample.

### MSC:

 26D05 Inequalities for trigonometric functions and polynomials 26C05 Real polynomials: analytic properties, etc. 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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### References:

  R. BENEDETTI and J.-J. RISLER, Real algebraic and semi-algebraic sets, Paris, Hermann, 1990. · Zbl 0694.14006  A. PARMEGGIANI, Subunit balls for symbols of pseudodifferential operators, Advances in Math., (to appear). · Zbl 0940.35214  A. ROBINSON, Introduction to model theory and the metamathematics of algebra, Amsterdam, North-Holland, 1963, revised 1974. · Zbl 0118.25302
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