static ex binomial_sym(const ex & x, const numeric & y)
{
if (y.is_integer()) {
if (y.is_nonneg_integer()) {
const unsigned N = y.to_int();
if (N == 0) return _ex1;
if (N == 1) return x;
ex t = x.expand();
for (unsigned i = 2; i <= N; ++i)
t = (t * (x + i - y - 1)).expand() / i;
return t;
} else
return _ex0;
}
return binomial(x, y).hold();
}
/**
* 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;
}
