Math_D
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 Sep24 awarded Autobiographer Jul2 awarded Curious Jun29 comment What is the algebraic normal form of $F(x,y,z)= Trace (\alpha x^{24}) + x^{312} + yz$? ANF is unique for $\mathbb F_2$ but I do not know actually it is true or not for $\mathbb F_5$. There is an example for $\mathbb F_2$ here, math.stackexchange.com/questions/109296/… However, $\mathbb F_{5^6}$ is too large to apply this method.. Jun29 comment What is the algebraic normal form of $F(x,y,z)= Trace (\alpha x^{24}) + x^{312} + yz$? I mean, it is simply the new representation of $F$ when it is defined from $\mathbb F_5^6$ to $\mathbb F_5$. $$F(x_1,x_2,x_3,x_4,y,z)=?$$ Jun29 comment What is the algebraic normal form of $F(x,y,z)= Trace (\alpha x^{24}) + x^{312} + yz$? We can regard $\mathbb F_{5^4}$ as the 4-dimensional vector space $\mathbb F_5^4$ over $\mathbb F_5$. Then, we can redefine the function $F$ from $\mathbb F_5^6$ to $\mathbb F_5$ and represent it as a polynomial with the form $$f(x)=\sum_{I \in P(N)}a_Ix^I,$$ where $P(N)$ denotes power set of $N=\left\{1,\cdots,6\right\}$ and $a_I \in \mathbb F_5$ Jun28 asked What is the algebraic normal form of $F(x,y,z)= Trace (\alpha x^{24}) + x^{312} + yz$? Oct28 accepted $\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,where $c \in \mathbb Z_9$ and $w=e^{2\pi i/9}$ Oct27 comment $\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,where $c \in \mathbb Z_9$ and $w=e^{2\pi i/9}$ This seems precisely true to me but considering the finite field case I doupted a little. Actually, in finite field case this equals to an expression which includes Gauss sum. The answer is always $p$, seems so perfect... Oct26 comment $\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,where $c \in \mathbb Z_9$ and $w=e^{2\pi i/9}$ Any useful referances to solve this are welcome and can be accepted as an answer. Oct26 comment $\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,where $c \in \mathbb Z_9$ and $w=e^{2\pi i/9}$ How to calculate the last sum easily? Oct26 comment $\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,where $c \in \mathbb Z_9$ and $w=e^{2\pi i/9}$ Actually, the problem is given not only for $Z_9$. The sum is over $\mahbb Z_{p^2}$ where $p$ is a prime number. But my aim is to understand for a small set first. I thought to split into two pieces as the sum over the multiplicative group and the other Oct26 asked $\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,where $c \in \mathbb Z_9$ and $w=e^{2\pi i/9}$ Oct14 accepted Solve $r=(p-1)+pr_1+p^2r_2$ for $r_1$ and $r_2$ when $r(p-1) \equiv 1$ (mod $p^3$) Oct14 comment Solve $r=(p-1)+pr_1+p^2r_2$ for $r_1$ and $r_2$ when $r(p-1) \equiv 1$ (mod $p^3$) Actually, for $r \in \mathbb Z_{p}$ $r\equiv p-1$ (mod $p$) and for $r \in \mathbb Z_{p^2}$ $r\equiv (p-1)+p(p-2)$ (mod $p^2$). However, for this case $r \neq (p-1)+p(p-2)+p(p-3)$ as we would expect Oct14 asked Solve $r=(p-1)+pr_1+p^2r_2$ for $r_1$ and $r_2$ when $r(p-1) \equiv 1$ (mod $p^3$) Oct14 comment Is $\sum_{c \notin \mathbb Z_q}\psi(c)-\sum_{c \notin \mathbb Z_q}\psi(c^n)$ an integer? thank you. But I did not clearly understand why it is integer if I take the sum over $\mathbb Z_q^{\times}$ Oct13 revised Is $\sum_{c \notin \mathbb Z_q}\psi(c)-\sum_{c \notin \mathbb Z_q}\psi(c^n)$ an integer? fixed Oct13 comment Is $\sum_{c \notin \mathbb Z_q}\psi(c)-\sum_{c \notin \mathbb Z_q}\psi(c^n)$ an integer? I fixed the question. It is changed now... Sorry about the syntax.. Oct13 comment Is $\sum_{c \notin \mathbb Z_q}\psi(c)-\sum_{c \notin \mathbb Z_q}\psi(c^n)$ an integer? I fixed the question. Sorry about that... Oct13 revised Is $\sum_{c \notin \mathbb Z_q}\psi(c)-\sum_{c \notin \mathbb Z_q}\psi(c^n)$ an integer? corrected