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In this paper, we introduce some new results on the study of Z q -equivariant planar polynomial vector fields. The main conclusions are as follows.(1) For the Z 2 -equivariant planar cubic systems having two elementary foci, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z 2 -equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme 6 6 is proved. (2) On the basis of mentioned work in (1), by considering the bifurcation of a global limit cycle from infinity, we show that under small Z 2 -equivariant cubic perturbations, such bi-center cubic system has at least 13 limit cycles with the scheme 1 6 6 , i.e., we obtain that the Hilbert number H (3) ≥ 13. (3) For the Z 5 -equivariant planar polynomial vector field of degree 5, we shown that such system has at least five symmetric centers if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z 5 -equivariant perturbations, the conclusion that perturbed system has

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BackgroundHepatocellular carcinoma (HCC) is the sixth-most common malignancy worldwide. Multiple previous studies have assessed the relationship between TM6SF2 gene polymorphism and the risk of developing HCC, with discrepant conclusions reached. To assess the association of TM6SF2 rs58542926 T/C gene polymorphism with liver cancer, we performed the current meta-analysis.MethodsThis study queried the MEDLINE, PubMed, EMBASE, and CENTRAL databases from inception to April 2019. Case-control studies assessing the relationship between TM6SF2 rs5854292 locus polymorphism and liver cancer were selected according to inclusion and exclusion criteria. The Stata 12.0 software was employed for data analysis.ResultsA total of 5 articles, encompassing 6873 patients, met inclusion criteria and were included in the meta-analysis. Statistical analysis showed that the TM6SF2 gene polymorphism was significantly associated with liver cancer in the allele contrast, dominant, recessive and over dominant models (T vs C, OR = 1.621, 95%CI 1.379–1.905; CT + TT vs CC. OR = 1.541, 95%CI 1.351–1.758; TT vs CT + CC, OR = 2.897, 95%CI 1.690–4.966; CC + TT vs TC, OR = 0.693, 95%CI 0.576–0.834). The Egger’s test revealed no significant publication bias.ConclusionThe present findings suggest a significant association of TM6SF2 gene polymorphism with HCC risk in the entire population studied.