On the Sharpness of fewnomial bound and the number of components of a fewnomial hypersurface

2007/01/24 by Bihan, Frederic, Rojas, J. Maurice, Sottile, Frank
#14P99 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0701667

Abstract

We show the existence of systems of n polynomial equations in n variables, with a total of n+k+1 distinct monomial terms, possessing [n/k+1]^k nondegenerate positive solutions. (Here, [x] is the integer part of a positive number x.) This shows that the recent upper bound of (e²+3)/4 2^\binomk2 n^k for the number of nondegenerate positive solutions is asymptotically sharp for fixed k and large n. We also adapt a method of Perrucci to show that there are fewer than (e²+3)/4 2^\binomk2 2ⁿ n^k connected components in a smooth hypersurface in the positive orthant of Rⁿ defined by a polynomial with n+k+1 monomials. Our results hold for polynomials with real exponents.

Related

• Bounds on the number of real solutions to polynomial equations
• New bounds for the number of connected components of fewnomial hypersurfaces
• The sharp bound for the number of real solutions to polynomial equation systems
• Fewnomial bounds for completely mixed polynomial systems
• Improved fewnomial upper bounds from Wronskians and dessins d'enfant