Bounds on the number of real solutions to polynomial equations

@misc{citeulike:1422836, abstract = {We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z^n of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is asymptotically sharp for k fixed and n large.}, archivePrefix = {arXiv}, author = {Bates, Daniel J. and Bihan, Fr\&\#xe9;d\&\#xe9;ric and Sottile, Frank}, citeulike-article-id = {1422836}, eprint = {0706.4134}, keywords = {algebraic\_geometry, fewnomials}, month = {Jun}, posted-at = {2007-06-29 16:07:23}, priority = {1}, title = {Bounds on the number of real solutions to polynomial equations}, url = {http://arxiv.org/abs/0706.4134}, year = {2007} }

TY - ELEC ID - citeulike:1422836 L3 - citeulike-article-id:1422836 N2 - We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z^n of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is asymptotically sharp for k fixed and n large. TI - Bounds on the number of real solutions to polynomial equations KW - algebraic_geometry KW - fewnomials AU - Bates, Daniel AU - Bihan, Frédéric AU - Sottile, Frank PY - 2007/Jun/28/ UR - http://arxiv.org/abs/0706.4134 ER -