# fq_default_poly.h – univariate polynomials over finite fields

## Types, macros and constants

type fq_default_poly_t

## Memory management

void fq_default_poly_init(fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Initialises `poly` for use, with context ctx, and setting its length to zero. A corresponding call to `fq_default_poly_clear()` must be made after finishing with the `fq_default_poly_t` to free the memory used by the polynomial.

void fq_default_poly_init2(fq_default_poly_t poly, slong alloc, const fq_default_ctx_t ctx)

  Initialises `poly` with space for at least `alloc` coefficients and sets the length to zero. The allocated coefficients are all set to zero. A corresponding call to `fq_default_poly_clear()` must be made after finishing with the `fq_default_poly_t` to free the memory used by the polynomial.

void fq_default_poly_realloc(fq_default_poly_t poly, slong alloc, const fq_default_ctx_t ctx)

  Reallocates the given polynomial to have space for `alloc` coefficients. If `alloc` is zero the polynomial is cleared and then reinitialised. If the current length is greater than `alloc` the polynomial is first truncated to length `alloc`.

void fq_default_poly_fit_length(fq_default_poly_t poly, slong len, const fq_default_ctx_t ctx)

  If `len` is greater than the number of coefficients currently allocated, then the polynomial is reallocated to have space for at least `len` coefficients. No data is lost when calling this function.

  The function efficiently deals with the case where `fit_length` is called many times in small increments by at least doubling the number of allocated coefficients when length is larger than the number of coefficients currently allocated.

void fq_default_poly_clear(fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Clears the given polynomial, releasing any memory used. It must be reinitialised in order to be used again.

void _fq_default_poly_set_length(fq_default_poly_t poly, slong len, const fq_default_ctx_t ctx)

  Set the length of `poly` to `len`.

void fq_default_poly_truncate(fq_default_poly_t poly, slong newlen, const fq_default_ctx_t ctx)

  Truncates the polynomial to length at most \(n\).

void fq_default_poly_set_trunc(fq_default_poly_t poly1, fq_default_poly_t poly2, slong newlen, const fq_default_ctx_t ctx)

  Sets `poly1` to `poly2` truncated to length \(n\).

void fq_default_poly_reverse(fq_default_poly_t output, const fq_default_poly_t input, slong m, const fq_default_ctx_t ctx)

  Sets `output` to the reverse of `input`, thinking of it as a polynomial of length `m`, notionally zero-padded if necessary). The length `m` must be non-negative, but there are no other restrictions. The output polynomial will be set to length `m` and then normalised.

## Polynomial parameters

slong fq_default_poly_degree(const fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Returns the degree of the polynomial `poly`.

slong fq_default_poly_length(const fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Returns the length of the polynomial `poly`.

## Randomisation

void fq_default_poly_randtest(fq_default_poly_t f, flint_rand_t state, slong len, const fq_default_ctx_t ctx)

  Sets \(f\) to a random polynomial of length at most `len` with entries in the field described by `ctx`.

void fq_default_poly_randtest_not_zero(fq_default_poly_t f, flint_rand_t state, slong len, const fq_default_ctx_t ctx)

  Same as `fq_default_poly_randtest` but guarantees that the polynomial is not zero.

void fq_default_poly_randtest_monic(fq_default_poly_t f, flint_rand_t state, slong len, const fq_default_ctx_t ctx)

  Sets \(f\) to a random monic polynomial of length `len` with entries in the field described by `ctx`.

void fq_default_poly_randtest_irreducible(fq_default_poly_t f, flint_rand_t state, slong len, const fq_default_ctx_t ctx)

  Sets \(f\) to a random monic, irreducible polynomial of length `len` with entries in the field described by `ctx`.

## Assignment and basic manipulation

void fq_default_poly_set(fq_default_poly_t poly1, const fq_default_poly_t poly2, const fq_default_ctx_t ctx)

  Sets the polynomial `poly1` to the polynomial `poly2`.

void fq_default_poly_set_fq_default(fq_default_poly_t poly, const fq_default_t c, const fq_default_ctx_t ctx)

  Sets the polynomial `poly` to `c`.

void fq_default_poly_swap(fq_default_poly_t op1, fq_default_poly_t op2, const fq_default_ctx_t ctx)

  Swaps the two polynomials `op1` and `op2`.

void fq_default_poly_zero(fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Sets `poly` to the zero polynomial.

void fq_default_poly_one(fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Sets `poly` to the constant polynomial \(1\).

void fq_default_poly_gen(fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Sets `poly` to the polynomial \(x\).

void fq_default_poly_make_monic(fq_default_poly_t rop, const fq_default_poly_t op, const fq_default_ctx_t ctx)

  Sets `rop` to `op`, normed to have leading coefficient 1.

void fq_default_poly_set_nmod_poly(fq_default_poly_t rop, const nmod_poly_t op, const fq_default_ctx_t ctx)

  Sets the polynomial `rop` to the polynomial `op`.

void fq_default_poly_set_fmpz_mod_poly(fq_default_poly_t rop, const fmpz_mod_poly_t op, const fq_default_ctx_t ctx)

  Sets the polynomial `rop` to the polynomial `op`.

void fq_default_poly_set_fmpz_poly(fq_default_poly_t rop, const fmpz_poly_t op, const fq_default_ctx_t ctx)

  Sets the polynomial `rop` to the polynomial `op`.

## Getting and setting coefficients

void fq_default_poly_get_coeff(fq_default_t x, const fq_default_poly_t poly, slong n, const fq_default_ctx_t ctx)

  Sets \(x\) to the coefficient of \(X^n\) in `poly`.

void fq_default_poly_set_coeff(fq_default_poly_t poly, slong n, const fq_default_t x, const fq_default_ctx_t ctx)

  Sets the coefficient of \(X^n\) in `poly` to \(x\).

void fq_default_poly_set_coeff_fmpz(fq_default_poly_t poly, slong n, const fmpz_t x, const fq_default_ctx_t ctx)

  Sets the coefficient of \(X^n\) in the polynomial to \(x\), assuming \(n \geq 0\).

## Comparison

int fq_default_poly_equal(const fq_default_poly_t poly1, const fq_default_poly_t poly2, const fq_default_ctx_t ctx)

  Returns nonzero if the two polynomials `poly1` and `poly2` are equal, otherwise returns zero.

int fq_default_poly_equal_trunc(const fq_default_poly_t poly1, const fq_default_poly_t poly2, slong n, const fq_default_ctx_t ctx)

  Notionally truncate `poly1` and `poly2` to length \(n\) and return nonzero if they are equal, otherwise return zero.

int fq_default_poly_is_zero(const fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Returns whether the polynomial `poly` is the zero polynomial.

int fq_default_poly_is_one(const fq_default_poly_t op, const fq_default_ctx_t ctx)

  Returns whether the polynomial `poly` is equal to the constant polynomial \(1\).

int fq_default_poly_is_gen(const fq_default_poly_t op, const fq_default_ctx_t ctx)

  Returns whether the polynomial `poly` is equal to the polynomial \(x\).

int fq_default_poly_is_unit(const fq_default_poly_t op, const fq_default_ctx_t ctx)

  Returns whether the polynomial `poly` is a unit in the polynomial ring \(\mathbf{F}_q[X]\), i.e. if it has degree \(0\) and is non-zero.

int fq_default_poly_equal_fq_default(const fq_default_poly_t poly, const fq_default_t c, const fq_default_ctx_t ctx)

  Returns whether the polynomial `poly` is equal the (constant) \(\mathbf{F}_q\) element `c`

## Addition and subtraction

void fq_default_poly_add(fq_default_poly_t res, const fq_default_poly_t poly1, const fq_default_poly_t poly2, const fq_default_ctx_t ctx)

  Sets `res` to the sum of `poly1` and `poly2`.

void fq_default_poly_add_si(fq_default_poly_t res, const fq_default_poly_t poly1, slong c, const fq_default_ctx_t ctx)

  Sets `res` to the sum of `poly1` and `c`.

void fq_default_poly_add_series(fq_default_poly_t res, const fq_default_poly_t poly1, const fq_default_poly_t poly2, slong n, const fq_default_ctx_t ctx)

  Notionally truncate `poly1` and `poly2` to length `n` and set `res` to the sum.

void fq_default_poly_sub(fq_default_poly_t res, const fq_default_poly_t poly1, const fq_default_poly_t poly2, const fq_default_ctx_t ctx)

  Sets `res` to the difference of `poly1` and `poly2`.

void fq_default_poly_sub_series(fq_default_poly_t res, const fq_default_poly_t poly1, const fq_default_poly_t poly2, slong n, const fq_default_ctx_t ctx)

  Notionally truncate `poly1` and `poly2` to length `n` and set `res` to the difference.

void fq_default_poly_neg(fq_default_poly_t res, const fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Sets `res` to the additive inverse of `poly`.

## Scalar multiplication and division

void fq_default_poly_scalar_mul_fq_default(fq_default_poly_t rop, const fq_default_poly_t op, const fq_default_t x, const fq_default_ctx_t ctx)

  Sets `rop` to the product of `op` by the scalar `x`, in the context defined by `ctx`.

void fq_default_poly_scalar_addmul_fq_default(fq_default_poly_t rop, const fq_default_poly_t op, const fq_default_t x, const fq_default_ctx_t ctx)

  Adds to `rop` the product of `op` by the scalar `x`, in the context defined by `ctx`.

void fq_default_poly_scalar_submul_fq_default(fq_default_poly_t rop, const fq_default_poly_t op, const fq_default_t x, const fq_default_ctx_t ctx)

  Subtracts from `rop` the product of `op` by the scalar `x`, in the context defined by `ctx`.

void fq_default_poly_scalar_div_fq_default(fq_default_poly_t rop, const fq_default_poly_t op, const fq_default_t x, const fq_default_ctx_t ctx)

  Sets `rop` to the quotient of `op` by the scalar `x`, in the context defined by `ctx`. An exception is raised if `x` is zero.

## Multiplication

void fq_default_poly_mul(fq_default_poly_t rop, const fq_default_poly_t op1, const fq_default_poly_t op2, const fq_default_ctx_t ctx)

  Sets `rop` to the product of `op1` and `op2`, choosing an appropriate algorithm.

void fq_default_poly_mullow(fq_default_poly_t rop, const fq_default_poly_t op1, const fq_default_poly_t op2, slong n, const fq_default_ctx_t ctx)

  Sets `rop` to the lowest \(n\) coefficients of the product of `op1` and `op2`.

void fq_default_poly_mulhigh(fq_default_poly_t res, const fq_default_poly_t poly1, const fq_default_poly_t poly2, slong start, const fq_default_ctx_t ctx)

  Computes the product of `poly1` and `poly2` and writes the coefficients from `start` onwards into the high coefficients of `res`, the remaining coefficients being arbitrary but reduced.

void fq_default_poly_mulmod(fq_default_poly_t res, const fq_default_poly_t poly1, const fq_default_poly_t poly2, const fq_default_poly_t f, const fq_default_ctx_t ctx)

  Sets `res` to the remainder of the product of `poly1` and `poly2` upon polynomial division by `f`.

## Squaring

void fq_default_poly_sqr(fq_default_poly_t rop, const fq_default_poly_t op, const fq_default_ctx_t ctx)

  Sets `rop` to the square of `op`, choosing an appropriate algorithm.

## Powering

void fq_default_poly_pow(fq_default_poly_t rop, const fq_default_poly_t op, ulong e, const fq_default_ctx_t ctx)

  Computes `rop = op^e`. If \(e\) is zero, returns one, so that in particular `0^0 = 1`.

void fq_default_poly_powmod_ui_binexp(fq_default_poly_t res, const fq_default_poly_t poly, ulong e, const fq_default_poly_t f, const fq_default_ctx_t ctx)

  Sets `res` to `poly` raised to the power `e` modulo `f`, using binary exponentiation. We require `e >= 0`.

void fq_default_poly_powmod_fmpz_binexp(fq_default_poly_t res, const fq_default_poly_t poly, const fmpz_t e, const fq_default_poly_t f, const fq_default_ctx_t ctx)

  Sets `res` to `poly` raised to the power `e` modulo `f`, using binary exponentiation. We require `e >= 0`.

void fq_default_poly_pow_trunc(fq_default_poly_t res, const fq_default_poly_t poly, ulong e, slong trunc, const fq_default_ctx_t ctx)

  Sets `res` to the low `trunc` coefficients of `poly` to the power `e`. This is equivalent to doing a powering followed by a truncation.

## Shifting

void fq_default_poly_shift_left(fq_default_poly_t rop, const fq_default_poly_t op, slong n, const fq_default_ctx_t ctx)

  Sets `rop` to `op` shifted left by \(n\) coeffs. Zero coefficients are inserted.

void fq_default_poly_shift_right(fq_default_poly_t rop, const fq_default_poly_t op, slong n, const fq_default_ctx_t ctx)

  Sets `rop` to `op` shifted right by \(n\) coefficients. If \(n\) is equal to or greater than the current length of `op`, `rop` is set to the zero polynomial.

## Norms

slong fq_default_poly_hamming_weight(const fq_default_poly_t op, const fq_default_ctx_t ctx)

  Returns the number of non-zero entries in the polynomial `op`.

## Euclidean division

void fq_default_poly_divrem(fq_default_poly_t Q, fq_default_poly_t R, const fq_default_poly_t A, const fq_default_poly_t B, const fq_default_ctx_t ctx)

  Computes \(Q\), \(R\) such that \(A = B Q + R\) with \(0 \leq \operatorname{len}(R) < \operatorname{len}(B)\).

  Assumes that the leading coefficient of \(B\) is invertible. This can be taken for granted the context is for a finite field, that is, when \(p\) is prime and \(f(X)\) is irreducible.

void fq_default_poly_rem(fq_default_poly_t R, const fq_default_poly_t A, const fq_default_poly_t B, const fq_default_ctx_t ctx)

  Sets `R` to the remainder of the division of `A` by `B` in the context described by `ctx`.

void fq_default_poly_inv_series(fq_default_poly_t Qinv, const fq_default_poly_t Q, slong n, const fq_default_ctx_t ctx)

  Given `Q` find `Qinv` such that `Q * Qinv` is `1` modulo \(x^n\). The constant coefficient of `Q` must be invertible modulo the modulus of `Q`. An exception is raised if this is not the case or if `n = 0`.

void fq_default_poly_div_series(fq_default_poly_t Q, const fq_default_poly_t A, const fq_default_poly_t B, slong n, const fq_default_ctx_t ctx)

  Set \(Q\) to the quotient of the series \(A\) by \(B\), thinking of the series as though they were of length \(n\). We assume that the bottom coefficient of \(B\) is invertible.

## Greatest common divisor

void fq_default_poly_gcd(fq_default_poly_t rop, const fq_default_poly_t op1, const fq_default_poly_t op2, const fq_default_ctx_t ctx)

  Sets `rop` to the greatest common divisor of `op1` and `op2`, using the either the Euclidean or HGCD algorithm. The GCD of zero polynomials is defined to be zero, whereas the GCD of the zero polynomial and some other polynomial \(P\) is defined to be \(P\). Except in the case where the GCD is zero, the GCD \(G\) is made monic.

void fq_default_poly_xgcd(fq_default_poly_t G, fq_default_poly_t S, fq_default_poly_t T, const fq_default_poly_t A, const fq_default_poly_t B, const fq_default_ctx_t ctx)

  Computes the GCD of \(A\) and \(B\). The GCD of zero polynomials is defined to be zero, whereas the GCD of the zero polynomial and some other polynomial \(P\) is defined to be \(P\). Except in the case where the GCD is zero, the GCD \(G\) is made monic.

  Polynomials `S` and `T` are computed such that `S*A + T*B = G`. The length of `S` will be at most `lenB` and the length of `T` will be at most `lenA`.

## Divisibility testing

int fq_default_poly_divides(fq_default_poly_t Q, const fq_default_poly_t A, const fq_default_poly_t B, const fq_default_ctx_t ctx)

  Returns \(1\) if \(B\) divides \(A\) exactly and sets \(Q\) to the quotient, otherwise returns \(0\).

  This function is currently unoptimised and provided for convenience only.

## Derivative

void fq_default_poly_derivative(fq_default_poly_t rop, const fq_default_poly_t op, const fq_default_ctx_t ctx)

  Sets `rop` to the derivative of `op`.

## Square root

void fq_default_poly_invsqrt_series(fq_default_poly_t g, const fq_default_poly_t h, slong n, fq_default_ctx_t ctx)

  Set \(g\) to the series expansion of \(1/\sqrt{h}\) to order \(O(x^n)\). It is assumed that \(h\) has constant term 1.

void fq_default_poly_sqrt_series(fq_default_poly_t g, const fq_default_poly_t h, slong n, fq_default_ctx_t ctx)

  Set \(g\) to the series expansion of \(\sqrt{h}\) to order \(O(x^n)\). It is assumed that \(h\) has constant term 1.

int fq_default_poly_sqrt(fq_default_poly_t s, const fq_default_poly_t p, fq_default_ctx_t mod)

  If \(p\) is a perfect square, sets \(s\) to a square root of \(p\) and returns 1. Otherwise returns 0.

## Evaluation

void fq_default_poly_evaluate_fq_default(fq_default_t rop, const fq_default_poly_t f, const fq_default_t a, const fq_default_ctx_t ctx)

  Sets `rop` to the value of \(f(a)\).

  As the coefficient ring \(\mathbf{F}_q\) is finite, Horner’s method is sufficient.

## Composition

void fq_default_poly_compose(fq_default_poly_t rop, const fq_default_poly_t op1, const fq_default_poly_t op2, const fq_default_ctx_t ctx)

  Sets `rop` to the composition of `op1` and `op2`. To be precise about the order of composition, denoting `rop`, `op1`, and `op2` by \(f\), \(g\), and \(h\), respectively, sets \(f(t) = g(h(t))\).

void fq_default_poly_compose_mod(fq_default_poly_t res, const fq_default_poly_t f, const fq_default_poly_t g, const fq_default_poly_t h, const fq_default_ctx_t ctx)

  Sets `res` to the composition \(f(g)\) modulo \(h\). We require that \(h\) is nonzero.

## Output

int fq_default_poly_fprint_pretty(FILE *file, const fq_default_poly_t poly, const char *x, const fq_default_ctx_t ctx)

  Prints the pretty representation of `poly` to the stream `file`, using the string `x` to represent the indeterminate.

  In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fq_default_poly_print_pretty(const fq_default_poly_t poly, const char *x, const fq_default_ctx_t ctx)

  Prints the pretty representation of `poly` to `stdout`, using the string `x` to represent the indeterminate.

  In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fq_default_poly_fprint(FILE *file, const fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Prints the pretty representation of `poly` to the stream `file`.

  In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fq_default_poly_print(const fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Prints the representation of `poly` to `stdout`.

  In case of success, returns a positive value. In case of failure, returns a non-positive value.

char *fq_default_poly_get_str(const fq_default_poly_t poly, const fq_default_ctx_t ctx)

  Returns the plain FLINT string representation of the polynomial `poly`.

char *fq_default_poly_get_str_pretty(const fq_default_poly_t poly, const char *x, const fq_default_ctx_t ctx)

  Returns a pretty representation of the polynomial `poly` using the null-terminated string `x` as the variable name

## Inflation and deflation

void fq_default_poly_inflate(fq_default_poly_t result, const fq_default_poly_t input, ulong inflation, const fq_default_ctx_t ctx)

  Sets `result` to the inflated polynomial \(p(x^n)\) where \(p\) is given by `input` and \(n\) is given by `inflation`.

void fq_default_poly_deflate(fq_default_poly_t result, const fq_default_poly_t input, ulong deflation, const fq_default_ctx_t ctx)

  Sets `result` to the deflated polynomial \(p(x^{1/n})\) where \(p\) is given by `input` and \(n\) is given by `deflation`. Requires \(n > 0\).

ulong fq_default_poly_deflation(const fq_default_poly_t input, const fq_default_ctx_t ctx)

  Returns the largest integer by which `input` can be deflated. As special cases, returns 0 if `input` is the zero polynomial and 1 of `input` is a constant polynomial.