Timeline for Sum of binary quadratic forms over inputs of equal Hamming weight
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 20, 2021 at 3:51 | comment | added | Max Alekseyev | In fact, $C_{n,a}$ can be similarly computed modulo $2^m$ for $m>2$, although formulas become cumbersome as $m$ grows. So, while in principle taking $2^m>\binom{n}a$ can reveal the value of $C_{q,a}$ itself, direct computation of $C_{q,a}$ will likely be easier. | |
| Aug 19, 2021 at 21:38 | comment | added | Max Alekseyev | My intuition only tells me that computing $C_{q,a}$ may be hard. | |
| Aug 19, 2021 at 20:05 | comment | added | Ben | Thank you! Do you have any intuition for the general question? I would also be interested if you know of a well-behaved, (hopefully "small") family of $\mathbb{Z}$-valued functions on $\{0,1,\dots, n\}$ that these $C_{q,\cdot}$ must lie in. | |
| Aug 19, 2021 at 14:44 | comment | added | Max Alekseyev | (sign corrected) We can compute $C_{q,a}\bmod4$ by considering $q(x)$ as a function over integers, and noticing that $(−1)^{q(x)}\equiv1+2q(x)\pmod4$. Then we have $$C_{q,a}\equiv \binom{n}{a} + 2\binom{n-1}{a-1}\sum_i \alpha_i + 2\binom{n-2}{a-2} \sum_{j<k} \beta_{j,k} \pmod{4}.$$ | |
| Aug 19, 2021 at 2:44 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
indices corrected
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| Aug 19, 2021 at 2:16 | comment | added | Max Alekseyev | Somewhat relevant paper: Algorithms for Modular Counting of Roots of Multivariate Polynomials | |
| Aug 18, 2021 at 18:00 | history | asked | Ben | CC BY-SA 4.0 |