padic.h

Returns a pointer to the unit part $u$, which can be passed to the functions of fmpz.h.

Return the valuation $v$ and the precision $N$ of op.

Initialises ctx for the prime $p$ with precomputed powers $p^e$ for $\mathtt{min} \le e < \mathtt{max}$ and the given printing mode. Throws unless $0 \le \mathtt{min} \le \mathtt{max}$. That $p$ is prime is not checked; for composite $p$ the behaviour is undefined.

Clears the context.

Sets rop to $p^e$, copying it from the table when possible. Throws if $e$ exceeds WORD_MAX.

Makes the uninitialised rop equal to $p^e$. If $p^e$ is in the table, rop becomes a shallow copy of the table entry, which must be neither modified nor cleared, and the function returns 0. Otherwise it initialises rop, computes the power and returns 1, and the caller must clear rop. Throws if $e$ exceeds WORD_MAX.

Initialise rop to zero with precision PADIC_DEFAULT_PREC (that is, 20), respectively $N$.

Clears rop.

Brings rop into canonical form by moving all factors $p$ of $u$ into $v$, and sets $v = 0$ if $u = 0$. Does not reduce.

Reduces the canonical rop modulo $p^N$: replaces $u$ by $u \bmod p^{N - v}$ if $v < N$, and sets rop to zero otherwise.

Canonicalises and reduces rop.

Sets rop to a random reduced element. Its valuation, when it is nonzero, lies in $[-\lceil N/10 \rceil, N)$ for $N > 0$, in $[N - \lceil -N/10 \rceil, N)$ for $N < 0$ and in $[-10, 0)$ for $N = 0$.

Like padic_randtest, but nonzero: after ten zero draws it sets rop to $p^{N - 1}$.

Sets rop to a random reduced $p$-adic integer, with valuation in $[0, N)$, or to zero if $N \le 0$.

Sets rop to op. The precision of rop is kept.

Set rop to the integer op.

Sets rop to the canonical rational number op. The power of $p$ in the denominator becomes a negative valuation, and the rest of the denominator is inverted modulo $p^{N - v}$.

Set rop to the GMP integer or rational op. These are declared only if gmp.h is included before padic.h.

Sets rop to the integer $p^v u$ represented by op. Throws if $v < 0$.

Sets rop to the rational number $p^v u$ represented by op.

GMP versions of padic_get_fmpz (which throws if $v < 0$) and padic_get_fmpq. Declared only if gmp.h is included before padic.h.

Swaps op1 and op2, including their precisions. This differs from an exchange through padic_set when the precisions differ.

Sets rop to zero.

Sets rop to one, or to zero if its precision is not positive.

Returns whether op is zero, that is, whether $u = 0$.

Returns whether $u = 1$ and $v = 0$.

Returns whether the stored values are equal, that is, whether $u_1 = u_2$ and $v_1 = v_2$. The precisions are not compared: elements of different precisions that agree modulo the smaller power of $p$ are equal only if their stored values coincide.

Set rop to $op_1 + op_2$ and $op_1 - op_2$, reduced.

Sets rop to $-op$, reduced.

Sets rop to $op_1 \, op_2$, reduced: the units are multiplied and the product is reduced modulo $p^{N - v_1 - v_2}$.

Sets rop to $op \cdot p^v$, reduced. Here $v$ may be negative.

Sets rop to $op_1 / op_2$, reduced. Throws if $op_2$ is zero.

Sets rop to $1/op$, reduced. For $op = p^v u$ this is $p^{-v}$ times the inverse of $u$ modulo $p^{N + v}$, computed by Hensel lifting; if $N + v \le 0$ the result is zero. Throws if op is zero.

Returns whether op is a square in $\mathbf{Q}_p$ and in that case sets rop to a square root, reduced. If the return value is 0, the value of rop is undefined.

Sets rop to $op^e$, reduced, for any integer $e$, with $op^0 = 1$. For $e < 0$ the unit is first inverted modulo $p^{N - ev}$. If op is zero and $e \ne 0$ the result is zero, also for $e < 0$. Assumes that $ev$ does not overflow a slong.

Returns $n$ such that $\operatorname{ord}_p(x^j / j!) \ge N$ for all $j \ge n$ and all $x$ with $\operatorname{ord}_p(x) = v$, so that the exponential series can be truncated after $n$ terms. Requires $1 \le v < N$ for odd $p$ and $2 \le v < N$ for $p = 2$.

Set rop to $\exp(p^v u)$ reduced modulo $p^N$. Require $p^v u \ne 0$, $v < N$, and $v \ge 1$ for odd $p$, $v \ge 2$ for $p = 2$. Allow aliasing between rop and $u$. _padic_exp uses the rectangular algorithm for $N < 1024$ and the balanced one otherwise.

Return whether the exponential series converges at op, that is, whether op is zero or has valuation at least 1 (odd $p$) or 2 ($p = 2$), and if so set rop to $\exp(op)$, reduced. Otherwise rop is not changed. The rectangular version evaluates the series by rectangular splitting, the balanced version by the $p$-adic analogue of the bit-burst method with binary splitting, which is quasi-linear in $N$ for fixed $p$.

Returns $b$ such that $iv - \operatorname{ord}_p(i) \ge N$ for all $i \ge b$, so that the logarithm series for an argument of valuation $v$ can be truncated after $b - 1$ terms. Requires $1 \le v < N$ and $N < 2^{\mathtt{FLINT\_BITS} - 2}$; throws if $N$ is larger.

Set $z$ to

for an integer $y \ne 0$ with $v = \operatorname{ord}_p(y)$ and $1 \le v < N$; the logarithm of $x$ is obtained with $y = 1 - x$. Do not allow aliasing between $y$ and $z$. _padic_log uses the rectangular version for $N < 512 / b$, where $b$ is the bit length of $p$, and the balanced version otherwise. The balanced version does not use $v$.

Return whether the logarithm series converges at op, that is, whether $\operatorname{ord}_p(op - 1) \ge 1$, and if so set rop to $\log(op)$, reduced. Otherwise rop is not changed. The condition is the same for every $p$, including $p = 2$. The rectangular version evaluates the series by rectangular splitting, the Satoh version first raises op to the power $p^k$ with $k = \lfloor \sqrt N \rfloor$, and the balanced version writes op as a product of factors whose logarithms are summed by binary splitting.

Sets rop to the Teichmüller lift of the $p$-adic unit op modulo $p^N$: the unique root of unity of order dividing $p - 1$ that is congruent to op modulo $p$. Requires $N \ge 1$ and allows aliasing. For $p = 2$ the result is 1.

Sets rop to the Teichmüller lift of op, reduced. If op is zero or divisible by $p$ the result is zero (which satisfies $t^p = t$ but is not a root of unity). Throws if op has negative valuation.

Return $\operatorname{ord}_2(n!)$ and $\operatorname{ord}_p(n!)$, by Legendre's formula $\sum_{k \ge 1} \lfloor n / p^k \rfloor$. The result fits in a ulong since $\operatorname{ord}_p(n!) \le (n - 1)/(p - 1)$. padic_val_fac_ui returns 0 if $p$ does not fit in a ulong.

Sets rop to $\operatorname{ord}_p(op!)$. Throws if op is negative.

Return the string representation of op in the given printing mode, respectively the mode of the context. If str is NULL, a new string is allocated with flint_malloc; otherwise str must be large enough and is returned.

Print $p^v u$, respectively op, to file or to stdout in the printing mode of the context, and return 1.

Prints the raw fields of op to stdout in the form (u v N).

Initialise S for inversion modulo $p^N$, $N \ge 1$, and clear it.

Sets rop to the inverse of the $p$-adic unit op modulo $p^N$, using the data S. The scratch space in S is overwritten, so S must not be shared between threads. Allows aliasing.

Sets rop to the inverse of the $p$-adic unit op modulo $p^N$, for $N \ge 1$. The result is in $[0, p^N)$. Allows aliasing.

Returns whether the $p$-adic unit op is a square in $\mathbf{Z}_p$ and if so sets rop to a square root modulo $p^N$, for $N \ge 1$. For odd $p$ this is decided modulo $p$, for $p = 2$ modulo 8. Allows aliasing.

Returns the sequence $a_0 = N$, $a_{i+1} = \lceil a_i/2 \rceil$, ending with $a_{n-1} = 1$, and sets *n to its length $\lceil \log_2 N \rceil + 1$. Requires $N \ge 1$. The array is allocated with flint_malloc and must be freed with flint_free.

Sets pow[i] to $p^{a_i}$ for a sequence $a$ of length $n$ as computed by _padic_lifts_exps, using about one multiplication per entry.