Some bounds for the number of components of real zero sets of sparse polynomials

D. PERRUCCI

DISCRETE AND COMPUTATIONAL GEOMETRY

Springer

Año: 2005 vol. 34 p. 475 - 495

0179-5376

We prove that the zero set of a 4-nomial in n variables in the positiveorthant has at most three connected components. This bound, whichdoes not depend on the degree of the polynomial, not only improves the best previously known bound (which was 10) but is optimal as well. In the general case, we prove that the number of connected components of the zero set of an m-nomial in n variables in the positive orthant is lower than or equal to (n+1)^(m-1) 2^(1+(m-1)(m-2)/2), improving slightly the known bounds. Finally, we show that for generic exponents, the number of non-compact connected components of the zero set of a 5-nomial in three variables in the positive octant is at most 12. This stronglyimproves the best previously known bound, which was 10384. All thebounds obtained in this paper continue to hold for real exponents.