All reports by Author Ashish Dwivedi:

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TR20-092
| 16th June 2020
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Ashish Dwivedi, Nitin Saxena#### Computing Igusa's local zeta function of univariates in deterministic polynomial-time

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TR19-033
| 20th February 2019
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Ashish Dwivedi, Rajat Mittal, Nitin Saxena#### Counting basic-irreducible factors mod $p^k$ in deterministic poly-time and $p$-adic applications

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TR19-008
| 20th January 2019
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Ashish Dwivedi, Rajat Mittal, Nitin Saxena#### Efficiently factoring polynomials modulo $p^4$

Ashish Dwivedi, Nitin Saxena

Igusa's local zeta function $Z_{f,p}(s)$ is the generating function that counts the number of integral roots, $N_{k}(f)$, of $f(\mathbf x) \bmod p^k$, for all $k$. It is a famous result, in analytic number theory, that $Z_{f,p}$ is a rational function in $\mathbb{Q}(p^s)$. We give an elementary proof of this fact ... more >>>

Ashish Dwivedi, Rajat Mittal, Nitin Saxena

Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that {\em counts} the number of irreducible factors of $f\bmod p$. We can ask the same question modulo prime-powers $p^k$. The irreducible ... more >>>

Ashish Dwivedi, Rajat Mittal, Nitin Saxena

Polynomial factoring has famous practical algorithms over fields-- finite, rational \& $p$-adic. However, modulo prime powers it gets hard as there is non-unique factorization and a combinatorial blowup ensues. For example, $x^2+p \bmod p^2$ is irreducible, but $x^2+px \bmod p^2$ has exponentially many factors! We present the first randomized poly($\deg ... more >>>