Problem:
Solution:
It is interesting to look at the finite field as boolean values and think of operators as logic gates. Multiply is AND and Addition is XOR, then this problem is really easy.
a) We have XOR(AND(x, x, y), AND(x, y, y)), there are two cases:
x and y are both 1, in this case, we got XOR(1, 1) = 0
otherwise, we get XOR(0, 0) = 0
Proposition 5 states that the only function that have every point in it zero set is the zero function if the field is infinite. In this case, the field is finite, that simply does not apply.
b) Using the same logic gate trick, we do this
$ g(x, y, z) = xyz + xyz^2 $
c) Using the very same trick, we also have
$ g(x_1, x_2, ... x_n) = x_1 x_2 ... x_n + x_1 x_2 ... x_n^2 $.
As a note, for any polynomial, we can add that polynomial to it to form another polynomial that yield exactly the same zero set!
Solution:
It is interesting to look at the finite field as boolean values and think of operators as logic gates. Multiply is AND and Addition is XOR, then this problem is really easy.
a) We have XOR(AND(x, x, y), AND(x, y, y)), there are two cases:
x and y are both 1, in this case, we got XOR(1, 1) = 0
otherwise, we get XOR(0, 0) = 0
Proposition 5 states that the only function that have every point in it zero set is the zero function if the field is infinite. In this case, the field is finite, that simply does not apply.
b) Using the same logic gate trick, we do this
$ g(x, y, z) = xyz + xyz^2 $
c) Using the very same trick, we also have
$ g(x_1, x_2, ... x_n) = x_1 x_2 ... x_n + x_1 x_2 ... x_n^2 $.
As a note, for any polynomial, we can add that polynomial to it to form another polynomial that yield exactly the same zero set!
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