Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let X be a semialgebraic set in Rn defined by a Boolean combination of atomic formulae of the kind h * 0 where * \in { >, \ge, = }, deg(h) < d, and the number of distinct polynomials h is k.  prove that the sum of Betti numbers of X is less than O(k2d)n.
['Andrei Gabrielov', 'Nicolai Vorobjov']
I am writing a research paper on a idea Betti Numbers of Semialgebraic Sets Defined by Quantifier-Free Formulae
[{"from": "human", "value": "I am writing a research paper on a idea Betti Numbers of Semialgebraic Sets Defined by Quantifier-Free Formulae"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let X be a semialgebraic set in Rn defined by a Boolean combination of atomic formulae of the kind h * 0 where * \\in { >, \\ge, = }, deg(h) < d, and the number of distinct polynomials h is k.  prove that the sum of Betti numbers of X is less than O(k2d)n."}]