Sign rule of Descartes

The sign rule of Descartes is in the math - similar to Sturm's theorem used to the maximum number of positive and negative - the zeros of a real polynomial to be determined.

rule

Descartes' rule of signs is:

The number of all positive zeros of a real polynomial is equal to the number of sign changes of its coefficient sequence or smaller than this by an even natural number, with each zero being counted according to its multiplicity.

The important conclusion is:

If a real polynomial has only one sign change, then it has exactly one single positive zero.

It is named after the French philosopher and mathematician René Descartes , who was the first to describe it in his work La Géométrie in 1637 .

Examples

Maximum number of positive zeros

With the polynomial

${\displaystyle f(x)=5x^{7}+6x^{6}-9x^{4}-4x^{3}-11x^{2}+7x-3}$

changes the sign of the coefficients three times. According to Descartes, the polynomial thus has three positive zeros or one positive zero. In fact, it has exactly one positive zero. ${\displaystyle f(x)}$

Maximum number of negative zeros

To determine the maximum number of negative zeros, a new polynomial is first created from the polynomial . This means that the signs of the coefficients are changed for an odd exponent , while the signs of the coefficients for an even exponent remain unchanged. Descartes' sign rule is then applied to this new polynomial. ${\displaystyle f(x)}$ ${\displaystyle f(-x)}$

If we look again at the polynomial

${\displaystyle f(x)=5x^{7}+6x^{6}-9x^{4}-4x^{3}-11x^{2}+7x-3}$

this is the new polynomial

${\displaystyle f(-x)=-5x^{7}+6x^{6}-9x^{4}+4x^{3}-11x^{2}-7x-3}$

Here the sign of the coefficients changes four times. According to Descartes, the polynomial has either four, two or no negative zeros. In fact, it doesn't have a negative zero. ${\displaystyle f(x)}$

literature

• Bruce Anderson, Jeffrey Jackson, Meera Sitharam: Descartes' Rule of Signs Revisited . In: American Mathematical Monthly , Vol. 105 (1998), pp. 447-451, ISSN  0002-9890 .
• David J. Grabiner: Descartes' Rule of Signs. Another construction . In: American Mathematical Monthly , Vol. 106 (1999), pp. 854-855, ISSN  0002-9890 .
• Henry S. Hall, Samuel R. Knight: Higher Algebra. A Sequel to Elementary Algebra for Schools . Maxford Books, New Delhi 2008, ISBN 81-8116-000-2 , pp. 450-460 (reprint of the London 1950 edition).
• Peter Henrici : Sign Changes. The Rule of Descartes . In: Power Series-Integration-Conformal-Mapping-Location of Zeros (Applied and Computational Complex Analysis; Vol. 1). Wiley, New York 1988, ISBN 0-471-60841-6 , pp. 439-443.
• Ilia Itenberg, Marie-Françoise Roy : Multivariate Descartes' Rule . In: Contributions to Algebra and Geometry , Vol. 37 (1996), No. 2, pp. 337-346.
• Oskar Perron : Algebra . De Gruyter, Berlin 1953, p. 17f. (Reprint of the Berlin 1933 edition).
• Dirk Struik (Ed.): A Source Book in Mathematics 1200-1800 . Princeton University Press, Princeton, NJ 1986, ISBN 0-691-08404-1 , pp. 89-93.