Marilyn Cameron

Marilyn Cameron

Answered

2022-10-25

Modular Polynomial Arithmetic
For p = 3 and n = 2, the 3 2 = 9 polynomials in the set are
0 x 2 x 1 x + 1 2 x + 1 2 x + 2 2 x + 2
For p = 2 and n = 3, the 2 3 = 8 polynomials in the set are
0 x + 1 x 2 + x 1 x 2 x 2 + x + 1 x x 2 + 1
For the first example right, from what I understand it is the polynomial whose coefficient are from Z p , { 0 , 1 , . . . , p 1 }. So given p = 3, I can get 0,1,2. And if given p = 2, I can get 0,1.
But I not sure how those x are derived. Any ideas?

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Answer & Explanation

Tania Alvarado

Tania Alvarado

Expert

2022-10-26Added 15 answers

Step 1
If n = 2, the polynomial has 2 terms, the x and constant term. If x = 3, the polynomial has three terms, x 2 , x, and the constant term.
Step 2
So the highest degree of the polynomial is n 1.

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gasavasiv

gasavasiv

Expert

2022-10-27Added 3 answers

Step 1
It seems to be the list of all polynomials of degree less than 2, with coefficients in Z/3Z in the first case; of all polynomials of degree less than 3, with coefficients in Z/2Z in the second case.
Indeed, for the first case, a polynomial of degree less than 2 has the form a 0 + a 1 x. Plug in the different possible values for the pairs ( a 0 , a 1 ), getting
( 0 , 0 ) 0 ( 0 , 1 ) x ( 0 , 2 ) 2 x ( 1 , 0 ) 1 ( 1 , 1 ) 1 + x ( 1 , 2 ) 1 + 2 x ( 2 , 0 ) 2 ( 2 , 1 ) 2 + x ( 2 , 2 ) 2 + 2 x
Step 2
The second case goe along the same lines:
( 0 , 0 , 1 ) ( 0 , 1 , 1 ) x 2 x + x 2 ( 0 , 0 , 0 ) ( 0 , 1 , 0 ) 0 x ( 1 , 0 , 1 ) ( 1 , 1 , 1 ) 1 + x 1 + x + x 2 ( 1 , 0 , 0 ) ( 1 , 1 , 0 ) 1 1 + x

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