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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
asked 22 hours ago
M. A. SARKAR
1,9461618
add a comment |
up vote
0
down vote
favorite
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
asked 22 hours ago
M. A. SARKAR
1,9461618
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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
asked 22 hours ago
M. A. SARKAR
1,9461618
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
asked 22 hours ago
M. A. SARKAR
1,9461618
asked 22 hours ago
M. A. SARKAR
1,9461618
edited 17 hours ago
asked 22 hours ago
M. A. SARKAR
1,9461618
asked 22 hours ago
M. A. SARKAR
1,9461618
asked 22 hours ago
M. A. SARKAR
1,9461618
1,9461618
add a comment |
add a comment |
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