/**
* Compute the primitive part and the content of a modular multivariate
* polynomial e \in Z_p[x_n][x_0, \ldots, x_{n-1}], i.e. e is considered
* a polynomial in variables x_0, \ldots, x_{n-1} with coefficients being
* modular polynomials Z_p[x_n]
* @param e polynomial to operate on
* @param pp primitive part of @a e, will be computed by this function
* @param c content (in the sense described above) of @a e, will be
* computed by this function
* @param vars variables x_0, \ldots, x_{n-1}, x_n
* @param p modulus
*/
void primpart_content(ex& pp, ex& c, ex e, const exvector& vars,
const long p)
{
static const ex ex1(1);
static const ex ex0(0);
e = e.expand();
if (e.is_zero()) {
pp = ex0;
c = ex1;
return;
}
exvector rest_vars = vars;
rest_vars.pop_back();
ex_collect_t ec;
collect_vargs(ec, e, rest_vars);
if (ec.size() == 1) {
// the input polynomial factorizes into
// p_1(x_n) p_2(x_0, \ldots, x_{n-1})
c = ec.rbegin()->second;
ec.rbegin()->second = ex1;
pp = ex_collect_to_ex(ec, rest_vars).expand().smod(numeric(p));
return;
}
// Start from the leading coefficient (which is stored as a last
// element of the terms array)
auto i = ec.rbegin();
ex g = i->second;
// there are at least two terms, so it's safe to...
++i;
while (i != ec.rend() && !g.is_equal(ex1)) {
g = euclid_gcd(i->second, g, vars.back(), p);
++i;
}
if (g.is_equal(ex1)) {
pp = e;
c = ex1;
return;
}
exvector mainvar;
mainvar.push_back(vars.back());
for (i = ec.rbegin(); i != ec.rend(); ++i) {
ex tmp(0);
if (!divide_in_z_p(i->second, g, tmp, mainvar, p))
throw std::logic_error(std::string(__func__) +
": bogus division failure");
i->second = tmp;
}
pp = ex_collect_to_ex(ec, rest_vars).expand().smod(numeric(p));
c = g;
}
/**
* Exact polynomial division of a, b \in Z_p[x_0, \ldots, x_n]
* It doesn't check whether the inputs are proper polynomials, so be careful
* of what you pass in.
*
* @param a first multivariate polynomial (dividend)
* @param b second multivariate polynomial (divisor)
* @param q quotient (returned)
* @param var variables X iterator to first element of vector of symbols
*
* @return "true" when exact division succeeds (the quotient is returned in
* q), "false" otherwise.
* @note @a p = 0 means the base ring is Z
*/
bool divide_in_z_p(const ex &a, const ex &b, ex &q, const exvector& vars, const long p)
{
static const ex _ex1(1);
if (b.is_zero())
throw(std::overflow_error("divide_in_z: division by zero"));
if (b.is_equal(_ex1)) {
q = a;
return true;
}
if (is_exactly_a<numeric>(a)) {
if (is_exactly_a<numeric>(b)) {
// p == 0 means division in Z
if (p == 0) {
const numeric tmp = ex_to<numeric>(a/b);
if (tmp.is_integer()) {
q = tmp;
return true;
} else
return false;
} else {
q = (a*recip(ex_to<numeric>(b), p)).smod(numeric(p));
return true;
}
} else
return false;
}
if (a.is_equal(b)) {
q = _ex1;
return true;
}
// Main symbol
const ex &x = vars.back();
// Compare degrees
int adeg = a.degree(x), bdeg = b.degree(x);
if (bdeg > adeg)
return false;
// Polynomial long division (recursive)
ex r = a.expand();
if (r.is_zero())
return true;
int rdeg = adeg;
ex eb = b.expand();
ex blcoeff = eb.coeff(x, bdeg);
exvector v;
v.reserve(std::max(rdeg - bdeg + 1, 0));
exvector rest_vars(vars);
rest_vars.pop_back();
while (rdeg >= bdeg) {
ex term, rcoeff = r.coeff(x, rdeg);
if (!divide_in_z_p(rcoeff, blcoeff, term, rest_vars, p))
break;
term = (term*power(x, rdeg - bdeg)).expand();
v.push_back(term);
r = (r - term*eb).expand();
if (p != 0)
r = r.smod(numeric(p));
if (r.is_zero()) {
q = (new add(v))->setflag(status_flags::dynallocated);
return true;
}
rdeg = r.degree(x);
}
return false;
}
