@article{Zhang2024, 
author = {Yang Zhang and Jizhu Nan},
title = {A note on the degree bounds of the invariant ring},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {5},
pages = {10869-10881},
keywords = {degree bounds, indecomposable module, symmetric algebra, periodicity property},
url = {https://www.sciopen.com/article/10.3934/math.2024530},
doi = {10.3934/math.2024530},
abstract = {Let    G  =      C    p    ×  H be a finite group, where        C    p   is a cyclic group of prime order    p and    H is a        p          ′      -group. Let        F   be an algebraically closed field in characteristic    p. Let    V be a direct sum of    m non-trivial indecomposable    G-modules such that the norm polynomials of the simple    H-modules are the power of the product of the basis elements of the dual. In previous work, we proved the periodicity property of the polynomial ring        F    [  V  ] with actions of    G. In this paper, by the periodicity property, we showed that        F    [  V      ]    G   is generated by    m norm polynomials together with homogeneous invariants of degree at most    m      |    G      |    −      d    i    m    (  V  ) and transfer invariants, which yields the well-known degree bound        d    i    m     (  V  )  ⋅  (      |    G      |    −  1  ). More precisely, we found that this bound gets less sharp as the dimensions of simple    H-modules increase.}
}