static void dsum_p(
fmpz_t rop,
const fmpz *dinv, const fmpz *mu, long M, const long *C, long lenC,
const fmpz_t a, long ui, long vi, long n, long d, long p, long N)
{
long m, r, idx;
fmpz_t apm1, apow, f, g, P, PN;
fmpz_init(apm1);
fmpz_init(apow);
fmpz_init(f);
fmpz_init(g);
fmpz_init_set_ui(P, p);
fmpz_init(PN);
fmpz_pow_ui(PN, P, N);
fmpz_zero(rop);
r = 0;
m = (p * (ui + 1) - (vi + 1)) / d;
if (m <= M) /* Step {r = 0} */
{
idx = _bsearch(C, 0, lenC, m % p);
fmpz_powm_ui(apm1, a, p - 1, PN);
fmpz_one(apow);
fmpz_one(f);
fmpz_mod(rop, mu + idx + lenC * (m / p), PN);
}
for (r = 1, m += p; m <= M; r++, m += p)
{
idx = _bsearch(C, 0, lenC, m % p);
fmpz_mul(apow, apow, apm1);
fmpz_mod(apow, apow, PN);
fmpz_mul_ui(f, f, ui + 1 + (r - 1) * d);
fmpz_mod(f, f, PN);
fmpz_mul(g, f, dinv + r);
fmpz_mul(g, g, apow);
fmpz_mul(g, g, mu + idx + lenC * (m / p));
fmpz_mod(g, g, PN);
fmpz_add(rop, rop, g);
}
fmpz_mod(rop, rop, PN);
fmpz_clear(apm1);
fmpz_clear(apow);
fmpz_clear(f);
fmpz_clear(g);
fmpz_clear(P);
fmpz_clear(PN);
}
void fmpz_powm(fmpz_t f, const fmpz_t g, const fmpz_t e, const fmpz_t m)
{
if (fmpz_sgn(m) <= 0)
{
flint_printf("Exception (fmpz_powm). Modulus is less than 1.\n");
abort();
}
else if (!COEFF_IS_MPZ(*e)) /* e is small */
{
fmpz_powm_ui(f, g, *e, m);
}
else /* e is large */
{
if (!COEFF_IS_MPZ(*m)) /* m is small */
{
ulong g1 = fmpz_fdiv_ui(g, *m);
mpz_t g2, m2;
__mpz_struct *mpz_ptr;
flint_mpz_init_set_ui(g2, g1);
flint_mpz_init_set_ui(m2, *m);
mpz_ptr = _fmpz_promote(f);
mpz_powm(mpz_ptr, g2, COEFF_TO_PTR(*e), m2);
mpz_clear(g2);
mpz_clear(m2);
_fmpz_demote_val(f);
}
else /* m is large */
{
if (!COEFF_IS_MPZ(*g)) /* g is small */
{
mpz_t g2;
__mpz_struct *mpz_ptr;
flint_mpz_init_set_si(g2, *g);
mpz_ptr = _fmpz_promote(f);
mpz_powm(mpz_ptr, g2, COEFF_TO_PTR(*e), COEFF_TO_PTR(*m));
mpz_clear(g2);
_fmpz_demote_val(f);
}
else /* g is large */
{
__mpz_struct *mpz_ptr = _fmpz_promote(f);
mpz_powm(mpz_ptr,
COEFF_TO_PTR(*g), COEFF_TO_PTR(*e), COEFF_TO_PTR(*m));
_fmpz_demote_val(f);
}
}
}
}
static void alpha(fmpz_t rop, const long *u, const long *v,
const fmpz *a, const fmpz *dinv, const fmpz **mu, long M, const long **C, const long *lenC,
long n, long d, long p, long N)
{
const long ku = diagfrob_k(u, n, d);
long i;
fmpz_t f, g, P, PN;
fmpz_init(f);
fmpz_init(g);
fmpz_init_set_ui(P, p);
fmpz_init(PN);
fmpz_pow_ui(PN, P, N);
fmpz_pow_ui(rop, P, ku);
fmpz_mod(rop, rop, PN);
for (i = 0; i <= n; i++)
{
long e = (p * (u[i] + 1) - (v[i] + 1)) / d;
fmpz_powm_ui(f, a + i, e, PN);
dsum(g, dinv, mu[i], M, C[i], lenC[i], a + i, u[i], v[i], n, d, p, N);
fmpz_mul(rop, rop, f);
fmpz_mul(rop, rop, g);
fmpz_mod(rop, rop, PN);
}
if (ku % 2 != 0 && !fmpz_is_zero(rop))
{
fmpz_sub(rop, PN, rop);
}
fmpz_clear(f);
fmpz_clear(g);
fmpz_clear(P);
fmpz_clear(PN);
}
void frob(const mpoly_t P, const ctx_t ctxFracQt,
const qadic_t t1, const qadic_ctx_t Qq,
prec_t *prec, const prec_t *prec_in,
int verbose)
{
const padic_ctx_struct *Qp = &Qq->pctx;
const fmpz *p = Qp->p;
const long a = qadic_ctx_degree(Qq);
const long n = P->n - 1;
const long d = mpoly_degree(P, -1, ctxFracQt);
const long b = gmc_basis_size(n, d);
long i, j, k;
/* Diagonal fibre */
padic_mat_t F0;
/* Gauss--Manin Connection */
mat_t M;
mon_t *bR, *bC;
fmpz_poly_t r;
/* Local solution */
fmpz_poly_mat_t C, Cinv;
long vC, vCinv;
/* Frobenius */
fmpz_poly_mat_t F;
long vF;
fmpz_poly_mat_t F1;
long vF1;
fmpz_poly_t cp;
clock_t c0, c1;
double c;
if (verbose)
{
printf("Input:\n");
printf(" P = "), mpoly_print(P, ctxFracQt), printf("\n");
printf(" p = "), fmpz_print(p), printf("\n");
printf(" t1 = "), qadic_print_pretty(t1, Qq), printf("\n");
printf("\n");
fflush(stdout);
}
/* Step 1 {M, r} *********************************************************/
c0 = clock();
mat_init(M, b, b, ctxFracQt);
fmpz_poly_init(r);
gmc_compute(M, &bR, &bC, P, ctxFracQt);
{
fmpz_poly_t t;
fmpz_poly_init(t);
fmpz_poly_set_ui(r, 1);
for (i = 0; i < M->m; i++)
for (j = 0; j < M->n; j++)
{
fmpz_poly_lcm(t, r, fmpz_poly_q_denref(
(fmpz_poly_q_struct *) mat_entry(M, i, j, ctxFracQt)));
fmpz_poly_swap(r, t);
}
fmpz_poly_clear(t);
}
c1 = clock();
c = (double) (c1 - c0) / CLOCKS_PER_SEC;
if (verbose)
{
printf("Gauss-Manin connection:\n");
printf(" r(t) = "), fmpz_poly_print_pretty(r, "t"), printf("\n");
printf(" Time = %f\n", c);
printf("\n");
fflush(stdout);
}
{
qadic_t t;
qadic_init2(t, 1);
fmpz_poly_evaluate_qadic(t, r, t1, Qq);
if (qadic_is_zero(t))
{
printf("Exception (deformation_frob).\n");
printf("The resultant r evaluates to zero (mod p) at t1.\n");
abort();
}
qadic_clear(t);
}
/* Precisions ************************************************************/
if (prec_in != NULL)
{
*prec = *prec_in;
}
else
{
deformation_precisions(prec, p, a, n, d, fmpz_poly_degree(r));
}
if (verbose)
{
printf("Precisions:\n");
printf(" N0 = %ld\n", prec->N0);
printf(" N1 = %ld\n", prec->N1);
printf(" N2 = %ld\n", prec->N2);
printf(" N3 = %ld\n", prec->N3);
printf(" N3i = %ld\n", prec->N3i);
printf(" N3w = %ld\n", prec->N3w);
printf(" N3iw = %ld\n", prec->N3iw);
printf(" N4 = %ld\n", prec->N4);
printf(" m = %ld\n", prec->m);
printf(" K = %ld\n", prec->K);
printf(" r = %ld\n", prec->r);
printf(" s = %ld\n", prec->s);
printf("\n");
fflush(stdout);
}
/* Initialisation ********************************************************/
padic_mat_init2(F0, b, b, prec->N4);
fmpz_poly_mat_init(C, b, b);
fmpz_poly_mat_init(Cinv, b, b);
fmpz_poly_mat_init(F, b, b);
vF = 0;
fmpz_poly_mat_init(F1, b, b);
vF1 = 0;
fmpz_poly_init(cp);
/* Step 2 {F0} ***********************************************************/
{
padic_ctx_t pctx_F0;
fmpz *t;
padic_ctx_init(pctx_F0, p, FLINT_MIN(prec->N4 - 10, 0), prec->N4, PADIC_VAL_UNIT);
t = _fmpz_vec_init(n + 1);
c0 = clock();
mpoly_diagonal_fibre(t, P, ctxFracQt);
diagfrob(F0, t, n, d, prec->N4, pctx_F0, 0);
padic_mat_transpose(F0, F0);
c1 = clock();
c = (double) (c1 - c0) / CLOCKS_PER_SEC;
if (verbose)
{
printf("Diagonal fibre:\n");
printf(" P(0) = {"), _fmpz_vec_print(t, n + 1), printf("}\n");
printf(" Time = %f\n", c);
printf("\n");
fflush(stdout);
}
_fmpz_vec_clear(t, n + 1);
padic_ctx_clear(pctx_F0);
}
/* Step 3 {C, Cinv} ******************************************************/
/*
Compute C as a matrix over Z_p[[t]]. A is the same but as a series
of matrices over Z_p. Mt is the matrix -M^t, and Cinv is C^{-1}^t,
the local solution of the differential equation replacing M by Mt.
*/
c0 = clock();
{
const long K = prec->K;
padic_mat_struct *A;
gmde_solve(&A, K, p, prec->N3, prec->N3w, M, ctxFracQt);
gmde_convert_soln(C, &vC, A, K, p);
for(i = 0; i < K; i++)
padic_mat_clear(A + i);
free(A);
}
c1 = clock();
c = (double) (c1 - c0) / CLOCKS_PER_SEC;
if (verbose)
{
printf("Local solution:\n");
printf(" Time for C = %f\n", c);
fflush(stdout);
}
c0 = clock();
{
const long K = (prec->K + (*p) - 1) / (*p);
mat_t Mt;
padic_mat_struct *Ainv;
mat_init(Mt, b, b, ctxFracQt);
mat_transpose(Mt, M, ctxFracQt);
mat_neg(Mt, Mt, ctxFracQt);
gmde_solve(&Ainv, K, p, prec->N3i, prec->N3iw, Mt, ctxFracQt);
gmde_convert_soln(Cinv, &vCinv, Ainv, K, p);
fmpz_poly_mat_transpose(Cinv, Cinv);
fmpz_poly_mat_compose_pow(Cinv, Cinv, *p);
for(i = 0; i < K; i++)
padic_mat_clear(Ainv + i);
free(Ainv);
mat_clear(Mt, ctxFracQt);
}
c1 = clock();
c = (double) (c1 - c0) / CLOCKS_PER_SEC;
if (verbose)
{
printf(" Time for C^{-1} = %f\n", c);
printf("\n");
fflush(stdout);
}
/* Step 4 {F(t) := C(t) F(0) C(t^p)^{-1}} ********************************/
/*
Computes the product C(t) F(0) C(t^p)^{-1} modulo (p^{N_2}, t^K).
This is done by first computing the unit part of the product
exactly over the integers modulo t^K.
*/
c0 = clock();
{
fmpz_t pN;
fmpz_poly_mat_t T;
fmpz_init(pN);
fmpz_poly_mat_init(T, b, b);
for (i = 0; i < b; i++)
{
/* Find the unique k s.t. F0(i,k) is non-zero */
for (k = 0; k < b; k++)
if (!fmpz_is_zero(padic_mat_entry(F0, i, k)))
break;
if (k == b)
{
printf("Exception (frob). F0 is singular.\n\n");
abort();
}
for (j = 0; j < b; j++)
{
fmpz_poly_scalar_mul_fmpz(fmpz_poly_mat_entry(T, i, j),
fmpz_poly_mat_entry(Cinv, k, j),
padic_mat_entry(F0, i, k));
}
}
fmpz_poly_mat_mul(F, C, T);
fmpz_poly_mat_truncate(F, prec->K);
vF = vC + padic_mat_val(F0) + vCinv;
/* Canonicalise (F, vF) */
{
long v = fmpz_poly_mat_ord_p(F, p);
if (v == LONG_MAX)
{
printf("ERROR (deformation_frob). F(t) == 0.\n");
abort();
}
else if (v > 0)
{
fmpz_pow_ui(pN, p, v);
fmpz_poly_mat_scalar_divexact_fmpz(F, F, pN);
vF = vF + v;
}
}
/* Reduce (F, vF) modulo p^{N2} */
fmpz_pow_ui(pN, p, prec->N2 - vF);
fmpz_poly_mat_scalar_mod_fmpz(F, F, pN);
fmpz_clear(pN);
fmpz_poly_mat_clear(T);
}
c1 = clock();
c = (double) (c1 - c0) / CLOCKS_PER_SEC;
if (verbose)
{
printf("Matrix for F(t):\n");
printf(" Time = %f\n", c);
printf("\n");
fflush(stdout);
}
/* Step 5 {G = r(t)^m F(t)} **********************************************/
c0 = clock();
{
fmpz_t pN;
fmpz_poly_t t;
fmpz_init(pN);
fmpz_poly_init(t);
fmpz_pow_ui(pN, p, prec->N2 - vF);
/* Compute r(t)^m mod p^{N2-vF} */
if (prec->denR == NULL)
{
fmpz_mod_poly_t _t;
fmpz_mod_poly_init(_t, pN);
fmpz_mod_poly_set_fmpz_poly(_t, r);
fmpz_mod_poly_pow(_t, _t, prec->m);
fmpz_mod_poly_get_fmpz_poly(t, _t);
fmpz_mod_poly_clear(_t);
}
else
{
/* TODO: We don't really need a copy */
fmpz_poly_set(t, prec->denR);
}
fmpz_poly_mat_scalar_mul_fmpz_poly(F, F, t);
fmpz_poly_mat_scalar_mod_fmpz(F, F, pN);
/* TODO: This should not be necessary? */
fmpz_poly_mat_truncate(F, prec->K);
fmpz_clear(pN);
fmpz_poly_clear(t);
}
c1 = clock();
c = (double) (c1 - c0) / CLOCKS_PER_SEC;
if (verbose)
{
printf("Analytic continuation:\n");
printf(" Time = %f\n", c);
printf("\n");
fflush(stdout);
}
/* Steps 6 and 7 *********************************************************/
if (a == 1)
{
/* Step 6 {F(1) = r(t_1)^{-m} G(t_1)} ********************************/
c0 = clock();
{
const long N = prec->N2 - vF;
fmpz_t f, g, t, pN;
fmpz_init(f);
fmpz_init(g);
fmpz_init(t);
fmpz_init(pN);
fmpz_pow_ui(pN, p, N);
/* f := \hat{t_1}, g := r(\hat{t_1})^{-m} */
_padic_teichmuller(f, t1->coeffs + 0, p, N);
if (prec->denR == NULL)
{
_fmpz_mod_poly_evaluate_fmpz(g, r->coeffs, r->length, f, pN);
fmpz_powm_ui(t, g, prec->m, pN);
}
else
{
_fmpz_mod_poly_evaluate_fmpz(t, prec->denR->coeffs, prec->denR->length, f, pN);
}
_padic_inv(g, t, p, N);
/* F1 := g G(\hat{t_1}) */
for (i = 0; i < b; i++)
for (j = 0; j < b; j++)
{
const fmpz_poly_struct *poly = fmpz_poly_mat_entry(F, i, j);
const long len = poly->length;
if (len == 0)
{
fmpz_poly_zero(fmpz_poly_mat_entry(F1, i, j));
}
else
{
fmpz_poly_fit_length(fmpz_poly_mat_entry(F1, i, j), 1);
_fmpz_mod_poly_evaluate_fmpz(t, poly->coeffs, len, f, pN);
fmpz_mul(fmpz_poly_mat_entry(F1, i, j)->coeffs + 0, g, t);
fmpz_mod(fmpz_poly_mat_entry(F1, i, j)->coeffs + 0,
fmpz_poly_mat_entry(F1, i, j)->coeffs + 0, pN);
_fmpz_poly_set_length(fmpz_poly_mat_entry(F1, i, j), 1);
_fmpz_poly_normalise(fmpz_poly_mat_entry(F1, i, j));
}
}
vF1 = vF;
fmpz_poly_mat_canonicalise(F1, &vF1, p);
fmpz_clear(f);
fmpz_clear(g);
fmpz_clear(t);
fmpz_clear(pN);
}
c1 = clock();
c = (double) (c1 - c0) / CLOCKS_PER_SEC;
if (verbose)
{
printf("Evaluation:\n");
printf(" Time = %f\n", c);
printf("\n");
fflush(stdout);
}
}
else
{
/* Step 6 {F(1) = r(t_1)^{-m} G(t_1)} ********************************/
c0 = clock();
{
const long N = prec->N2 - vF;
fmpz_t pN;
fmpz *f, *g, *t;
fmpz_init(pN);
f = _fmpz_vec_init(a);
g = _fmpz_vec_init(2 * a - 1);
t = _fmpz_vec_init(2 * a - 1);
fmpz_pow_ui(pN, p, N);
/* f := \hat{t_1}, g := r(\hat{t_1})^{-m} */
_qadic_teichmuller(f, t1->coeffs, t1->length, Qq->a, Qq->j, Qq->len, p, N);
if (prec->denR == NULL)
{
fmpz_t e;
fmpz_init_set_ui(e, prec->m);
_fmpz_mod_poly_compose_smod(g, r->coeffs, r->length, f, a,
Qq->a, Qq->j, Qq->len, pN);
_qadic_pow(t, g, a, e, Qq->a, Qq->j, Qq->len, pN);
fmpz_clear(e);
}
else
{
_fmpz_mod_poly_reduce(prec->denR->coeffs, prec->denR->length, Qq->a, Qq->j, Qq->len, pN);
_fmpz_poly_normalise(prec->denR);
_fmpz_mod_poly_compose_smod(t, prec->denR->coeffs, prec->denR->length, f, a,
Qq->a, Qq->j, Qq->len, pN);
}
_qadic_inv(g, t, a, Qq->a, Qq->j, Qq->len, p, N);
/* F1 := g G(\hat{t_1}) */
for (i = 0; i < b; i++)
for (j = 0; j < b; j++)
{
const fmpz_poly_struct *poly = fmpz_poly_mat_entry(F, i, j);
const long len = poly->length;
fmpz_poly_struct *poly2 = fmpz_poly_mat_entry(F1, i, j);
if (len == 0)
{
fmpz_poly_zero(poly2);
}
else
{
_fmpz_mod_poly_compose_smod(t, poly->coeffs, len, f, a,
Qq->a, Qq->j, Qq->len, pN);
fmpz_poly_fit_length(poly2, 2 * a - 1);
_fmpz_poly_mul(poly2->coeffs, g, a, t, a);
_fmpz_mod_poly_reduce(poly2->coeffs, 2 * a - 1, Qq->a, Qq->j, Qq->len, pN);
_fmpz_poly_set_length(poly2, a);
_fmpz_poly_normalise(poly2);
}
}
/* Now the matrix for p^{-1} F_p at t=t_1 is (F1, vF1). */
vF1 = vF;
fmpz_poly_mat_canonicalise(F1, &vF1, p);
fmpz_clear(pN);
_fmpz_vec_clear(f, a);
_fmpz_vec_clear(g, 2 * a - 1);
_fmpz_vec_clear(t, 2 * a - 1);
}
c1 = clock();
c = (double) (c1 - c0) / CLOCKS_PER_SEC;
if (verbose)
{
printf("Evaluation:\n");
printf(" Time = %f\n", c);
printf("\n");
fflush(stdout);
}
/* Step 7 {Norm} *****************************************************/
/*
Computes the matrix for $q^{-1} F_q$ at $t = t_1$ as the
product $F \sigma(F) \dotsm \sigma^{a-1}(F)$ up appropriate
transpositions because our convention of columns vs rows is
the opposite of that used by Gerkmann.
Note that, in any case, transpositions do not affect
the characteristic polynomial.
*/
c0 = clock();
{
const long N = prec->N1 - a * vF1;
fmpz_t pN;
fmpz_poly_mat_t T;
fmpz_init(pN);
fmpz_poly_mat_init(T, b, b);
fmpz_pow_ui(pN, p, N);
fmpz_poly_mat_frobenius(T, F1, 1, p, N, Qq);
_qadic_mat_mul(F1, F1, T, pN, Qq);
for (i = 2; i < a; i++)
{
fmpz_poly_mat_frobenius(T, T, 1, p, N, Qq);
_qadic_mat_mul(F1, F1, T, pN, Qq);
}
vF1 = a * vF1;
fmpz_poly_mat_canonicalise(F1, &vF1, p);
fmpz_clear(pN);
fmpz_poly_mat_clear(T);
}
c1 = clock();
c = (double) (c1 - c0) / CLOCKS_PER_SEC;
if (verbose)
{
printf("Norm:\n");
printf(" Time = %f\n", c);
printf("\n");
fflush(stdout);
}
}
/* Step 8 {Reverse characteristic polynomial} ****************************/
c0 = clock();
deformation_revcharpoly(cp, F1, vF1, n, d, prec->N0, prec->r, prec->s, Qq);
c1 = clock();
c = (double) (c1 - c0) / CLOCKS_PER_SEC;
if (verbose)
{
printf("Reverse characteristic polynomial:\n");
printf(" p(T) = "), fmpz_poly_print_pretty(cp, "T"), printf("\n");
printf(" Time = %f\n", c);
printf("\n");
fflush(stdout);
}
/* Clean up **************************************************************/
padic_mat_clear(F0);
mat_clear(M, ctxFracQt);
free(bR);
free(bC);
fmpz_poly_clear(r);
fmpz_poly_mat_clear(C);
fmpz_poly_mat_clear(Cinv);
fmpz_poly_mat_clear(F);
fmpz_poly_mat_clear(F1);
fmpz_poly_clear(cp);
}
void _qadic_norm_resultant(fmpz_t rop, const fmpz *op, slong len,
const fmpz *a, const slong *j, slong lena,
const fmpz_t p, slong N)
{
const slong d = j[lena - 1];
fmpz_t pN;
fmpz_init(pN);
fmpz_pow_ui(pN, p, N);
if (len == 1)
{
fmpz_powm_ui(rop, op + 0, d, pN);
}
else /* len >= 2 */
{
{
const slong n = d + len - 1;
slong i, k;
fmpz *M;
M = flint_calloc(n * n, sizeof(fmpz));
for (k = 0; k < len-1; k++)
{
for (i = 0; i < lena; i++)
{
M[k * n + k + (d - j[i])] = a[i];
}
}
for (k = 0; k < d; k++)
{
for (i = 0; i < len; i++)
{
M[(len-1 + k) * n + k + (len-1 - i)] = op[i];
}
}
_fmpz_mod_mat_det(rop, M, n, pN);
flint_free(M);
}
/*
XXX: This part of the code is currently untested as the Conway
polynomials used for the extension Qq/Qp are monic.
*/
if (!fmpz_is_one(a + (lena - 1)))
{
fmpz_t f;
fmpz_init(f);
fmpz_powm_ui(f, a + (lena - 1), len - 1, pN);
_padic_inv(f, f, p, N);
fmpz_mul(rop, f, rop);
fmpz_mod(rop, rop, pN);
fmpz_clear(f);
}
}
fmpz_clear(pN);
}
void precompute_muex(fmpz **mu, long M,
const long **C, const long *lenC,
const fmpz *a, long n, long p, long N)
{
const long ve = (p == 2) ? M / 4 + 1 : M / (p * (p - 1)) + 1;
fmpz_t P, pNe, pe;
fmpz_t apow, f, g, h;
fmpz *nu;
long *v;
long i, j;
fmpz_init_set_ui(P, p);
fmpz_init(pNe);
fmpz_init(pe);
fmpz_pow_ui(pNe, P, N + ve);
fmpz_pow_ui(pe, P, ve);
fmpz_init(apow);
fmpz_init(f);
fmpz_init(g);
fmpz_init(h);
/* Precompute $(l!)^{-1}$ */
nu = _fmpz_vec_init(M + 1);
v = malloc((M + 1) * sizeof(long));
{
long *D, lenD = 0, k = 0;
for (i = 0; i <= n; i++)
lenD += lenC[i];
D = malloc(lenD * sizeof(long));
for (i = 0; i <= n; i++)
for (j = 0; j < lenC[i]; j++)
D[k++] = C[i][j];
_remove_duplicates(D, &lenD);
_sort(D, lenD);
precompute_nu(nu, v, M, D, lenD, p, N + ve);
free(D);
}
for (i = 0; i <= n; i++)
{
long m = -1, quo, idx, w;
fmpz *z;
/* Set apow = a[i]^{-(p-1)} mod p^N */
fmpz_invmod(apow, a + i, pNe);
fmpz_powm_ui(apow, apow, p - 1, pNe);
/*
Run over all relevant m in [0, M].
Note that lenC[i] > 0 for all i.
*/
for (quo = 0; m <= M; quo++)
{
for (idx = 0; idx < lenC[i]; idx++)
{
m = quo * p + C[i][idx];
if (m > M)
break;
/*
Note that $\mu_m$ is equal to
$\sum_{k=0}^{\floor{m/p}} p^{\floor{m/p}-k}\nu_{m-pk}\nu_k$
where $\nu_i$ denotes the number with unit part nu[i]
and valuation v[i].
*/
w = (p == 2) ? (3 * m) / 4 - (m == 3 || m == 7) : m / p;
z = mu[i] + lenC[i] * quo + idx;
fmpz_zero(z);
fmpz_one(h);
for (j = 0; j <= m / p; j++)
{
fmpz_pow_ui(f, P, ve + w - j + v[m - p*j] + v[j]);
fmpz_mul(g, nu + (m - p*j), nu + j);
fmpz_mul(f, f, g);
fmpz_mul(f, f, h);
fmpz_add(z, z, f);
fmpz_mod(z, z, pNe);
/* Set h = a[i]^{- (j+1)(p-1)} mod p^{N+e} */
fmpz_mul(h, h, apow);
fmpz_mod(h, h, pNe);
}
fmpz_divexact(z, z, pe);
}
}
}
fmpz_clear(P);
fmpz_clear(pNe);
fmpz_clear(pe);
fmpz_clear(apow);
fmpz_clear(f);
fmpz_clear(g);
fmpz_clear(h);
_fmpz_vec_clear(nu, M + 1);
free(v);
}
