I got stuch finding Newton polygon of the following product with any easiest method











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Find the Newton polygon of the following polynomials:



$(i) f(X)=(1-X)(1-pX)(1-p^3X)$,



$ (ii) g(X)=prod_{i=1}^{p^2} (1-iX)$.



Answer:



$(i)$



To find the Newton polygon for the polynomial $f(X)$, we can simply multiply the linear factors as follows:



$ f(X)=(1-X)(1-pX)(1-p^3X) \ Rightarrow f(X)=1-(p^3+p+1)X+(p^4+p^3+p)X^2-p^4 X^3$.



Thus the Newton polygon has following vertices:



$ (0, ord_p(1)), (1, ord_p(-p^3+p+1)), (2, ord_p(p^4+p^3+p)) , (3,ord_p(p^4))$



i.e., $ (0,0), (1, 0), (2,1), (3,4)$.



But this process we can not apply to the second polynomial $g(X)$ in (ii) because it will be complicated.



So there should be other easy method to find the Newton polygon for the polynomials in $ (i)$ and $(ii)$.



Please help me find the Newton polygon for $(ii)$.






p-adic-number-theory local-field







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    Find the Newton polygon of the following polynomials:



    $(i) f(X)=(1-X)(1-pX)(1-p^3X)$,



    $ (ii) g(X)=prod_{i=1}^{p^2} (1-iX)$.



    Answer:



    $(i)$



    To find the Newton polygon for the polynomial $f(X)$, we can simply multiply the linear factors as follows:



    $ f(X)=(1-X)(1-pX)(1-p^3X) \ Rightarrow f(X)=1-(p^3+p+1)X+(p^4+p^3+p)X^2-p^4 X^3$.



    Thus the Newton polygon has following vertices:



    $ (0, ord_p(1)), (1, ord_p(-p^3+p+1)), (2, ord_p(p^4+p^3+p)) , (3,ord_p(p^4))$



    i.e., $ (0,0), (1, 0), (2,1), (3,4)$.



    But this process we can not apply to the second polynomial $g(X)$ in (ii) because it will be complicated.



    So there should be other easy method to find the Newton polygon for the polynomials in $ (i)$ and $(ii)$.



    Please help me find the Newton polygon for $(ii)$.






    p-adic-number-theory local-field







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      Find the Newton polygon of the following polynomials:



      $(i) f(X)=(1-X)(1-pX)(1-p^3X)$,



      $ (ii) g(X)=prod_{i=1}^{p^2} (1-iX)$.



      Answer:



      $(i)$



      To find the Newton polygon for the polynomial $f(X)$, we can simply multiply the linear factors as follows:



      $ f(X)=(1-X)(1-pX)(1-p^3X) \ Rightarrow f(X)=1-(p^3+p+1)X+(p^4+p^3+p)X^2-p^4 X^3$.



      Thus the Newton polygon has following vertices:



      $ (0, ord_p(1)), (1, ord_p(-p^3+p+1)), (2, ord_p(p^4+p^3+p)) , (3,ord_p(p^4))$



      i.e., $ (0,0), (1, 0), (2,1), (3,4)$.



      But this process we can not apply to the second polynomial $g(X)$ in (ii) because it will be complicated.



      So there should be other easy method to find the Newton polygon for the polynomials in $ (i)$ and $(ii)$.



      Please help me find the Newton polygon for $(ii)$.






      p-adic-number-theory local-field







      share|cite|improve this question















      Find the Newton polygon of the following polynomials:



      $(i) f(X)=(1-X)(1-pX)(1-p^3X)$,



      $ (ii) g(X)=prod_{i=1}^{p^2} (1-iX)$.



      Answer:



      $(i)$



      To find the Newton polygon for the polynomial $f(X)$, we can simply multiply the linear factors as follows:



      $ f(X)=(1-X)(1-pX)(1-p^3X) \ Rightarrow f(X)=1-(p^3+p+1)X+(p^4+p^3+p)X^2-p^4 X^3$.



      Thus the Newton polygon has following vertices:



      $ (0, ord_p(1)), (1, ord_p(-p^3+p+1)), (2, ord_p(p^4+p^3+p)) , (3,ord_p(p^4))$



      i.e., $ (0,0), (1, 0), (2,1), (3,4)$.



      But this process we can not apply to the second polynomial $g(X)$ in (ii) because it will be complicated.



      So there should be other easy method to find the Newton polygon for the polynomials in $ (i)$ and $(ii)$.



      Please help me find the Newton polygon for $(ii)$.






      p-adic-number-theory local-field




      p-adic-number-theory local-field






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      edited 17 hours ago

























      asked 22 hours ago









      M. A. SARKAR

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