HDU6414 带劲的多项式

Sep 12, 2019 由 小羊

考虑形如

$f(x) = \prod_{i=1}^r (x-\lambda_i)^{l_i}$

形式的多项式，它的一阶导函数为

$f'(x) = \left( \sum_{i=1}^{r}l_i(x-\lambda_i) \right) \left(\prod_{i=1}^r (x-\lambda_i)^{l_i -1}\right)$

则可以发现它们之间有

$\gcd(f(x), f'(x)) = \prod_{i=1}^r (x-\lambda_i)^{l_i-1}$

则有

$\frac{f(x)}{\gcd(f(x), f'(x))} = \prod_{i=1}^r (x-\lambda_i)$

由于根重数不同，每个根的因子会在不同时刻从上式消失。每次将 $\gcd(f, f')$ 当作新的 $f$ 进行迭代即可。

另外，STL好慢啊233。

#include <bits/stdc++.h>
using namespace std;
typedef long long lld;
const int MOD = 998244353;
typedef vector<int> Poly;

inline lld fpow(lld a, int k)
{
    lld b = 1;
    for (; k; k>>=1, a=a*a%MOD)
        if (k&1) b=b*a%MOD;
    return b;
}

void norm(Poly &a)
{
    lld qwq = fpow(a.back(), MOD-2);
    for (auto &x : a) x = x * qwq % MOD;
}

Poly diff(const Poly &a)
{
    Poly _a = Poly(a.size() - 1);
    for (int i = 0; i < _a.size(); i++)
        _a[i] = a[i+1] * (i+1LL) % MOD;
    return _a;
}

void divide(const Poly &a, const Poly &p, Poly &q, Poly &r)
{
    assert(p.size() >= 1 && p.back() != 0);
    q.clear(), r = a; int P = p.size();
    lld pq = fpow(p.back(), MOD-2);

    for (int i = a.size()-1; i >= P-1; i--)
    {
        q.push_back(r[i] * pq % MOD);
        for (int j = 0; j < P; j++)
            r[i-P+1+j] = (r[i-P+1+j] + MOD - 1LL * r[i] * pq % MOD * p[j] % MOD) % MOD;
    }

    reverse(q.begin(), q.end());
    while (q.back() == 0) q.pop_back();
    while (r.back() == 0) r.pop_back();
}

Poly gcd(Poly a, Poly b)
{
    Poly c, d;
    while (b.size()) divide(a, b, c, d), a = b, b = d;
    norm(a);
    return a;
}

void solve()
{
    int n; scanf("%d", &n); vector<pair<int,int>> ans;
    Poly orig(n+1, 0), dif, cbs, tmp, nil, nul;
    for (int i = 0; i <= n; i++) scanf("%d", &orig[i]);
    norm(orig); dif = diff(orig), tmp = gcd(orig, dif);
    divide(orig, tmp, cbs, nil); orig = tmp, norm(orig);

    for (int l = 1; dif.size(); l++)
    {
        dif = diff(orig), tmp = gcd(orig, dif);
        divide(orig, tmp, nul, nil);

        if (nul != cbs)
        {
            // an root appears!
            Poly a, b;
            divide(cbs, nul, b, a);
            assert(b.size() == 2);
            ans.emplace_back((MOD-b[0])%MOD, l);
            cbs = nul;
        }

        orig = tmp, norm(orig);
    }

    printf("%d\n", int(ans.size()));
    for (auto p : ans) printf("%d %d\n", p.first, p.second);
}

int main()
{
    int T; scanf("%d", &T);
    while (T--) solve();
    return 0;
}