Volume 10, Number 4, 2020, Pages 15341544 DOI：10.11948/20190247 
A linear estimation to the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one 
Lijun Hong,Xiaochun Hong,Junliang Lu 
Keywords:Abelian integral, quadratic reversible center, weakened Hilbert''s 16th problem, limit cycle. 
Abstract: 
In this paper, using the method of PicardFuchs equation and Riccati equation, we consider the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one under arbitrary polynomial perturbations of degree $n$, and obtain that the upper bound of the number is $2\left[{(n+1)}/{2}\right]+$ $\left[{n}/{2}\right]+2$ ($n\geq 1$), which linearly depends on $n$. 
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