Monomial orderings and exponent packing

Every sparse multivariate module of FLINT (fmpz_mpoly, fmpq_mpoly, nmod_mpoly, fmpz_mod_mpoly, fq_nmod_mpoly, fq_zech_mpoly and the generic gr_mpoly) uses the same distributed representation: a polynomial is a list of terms $c_i x^{e_i}$ with nonzero coefficients, sorted so that $x^{e_0} > x^{e_1} > \cdots$ for a fixed monomial ordering, with the coefficients in one array and the exponent vectors in another. The exponent vectors are packed: several exponents share a machine word, in a layout chosen so that the monomial operations needed by the algorithms become a few word operations. This design follows Monagan and Pearce [MonPea2007], who showed that packing is a main factor in the speed of sparse polynomial arithmetic, and Bachmann and Schönemann [BacSch1998] for the treatment of orderings. The functions are in the module mpoly.h, and the notes of Schultz [Sch-mpoly] give further details.

Monomial orderings#

A monomial ordering is a total order $<$ on the monomials $x^e = x_0^{e_0} \cdots x_{n-1}^{e_{n-1}}$, $e \in \mathbb{N}^n$, such that $1 \le x^e$ and $x^a < x^b \Rightarrow x^{a+c} < x^{b+c}$. Such an order is a well-order (Dickson's lemma), which is what makes the division algorithm and the heap algorithms terminate [CLO2015, §2.2]. FLINT supports three, given by ordering_t, with $|e| = e_0 + \cdots + e_{n-1}$:

• ORD_LEX: $x^a > x^b$ if the first nonzero entry of $a - b$ is positive;
• ORD_DEGLEX: $x^a > x^b$ if $|a| > |b|$, or $|a| = |b|$ and $x^a > x^b$ in lex order;
• ORD_DEGREVLEX: $x^a > x^b$ if $|a| > |b|$, or $|a| = |b|$ and the last nonzero entry of $a - b$ is negative.

The variable $x_0$ is the most significant one. The leading term $\operatorname{lt}(A)$, monomial $\operatorname{lm}(A)$ and coefficient $\operatorname{lc}(A)$ refer to the first term in this order. The GCD and factoring code converts its inputs to ORD_LEX, in which the leading coefficient with respect to $x_0$ is a polynomial in the other variables, as the recursive view $R[x_1, \ldots][x_0]$ requires.

Packed monomials#

Fix a field width $b$. The exponent vector $e$ is stored as a sequence of fields $f_0, f_1, \ldots, f_{k-1}$, each a $b$-bit unsigned integer, where $k = n$ for ORD_LEX and $k = n + 1$ for the degree orderings:

• for ORD_LEX and ORD_DEGLEX, field $n - 1 - i$ holds $e_i$, so that $e_0$ is in the most significant position among the variables;
• for ORD_DEGREVLEX, field $i$ holds $e_i$;
• for the degree orderings, field $n$ holds the total degree $|e|$.

The fields are concatenated, field 0 in the least significant bits, into the integer

which is stored in $N$ words, least significant word first. Two layouts are used:

• single precision, $8 \le b \le$ FLINT_BITS: a word holds $\lfloor \mathrm{FLINT\_BITS}/b \rfloor$ fields, fields do not straddle words, and unused high bits of a word are zero; $N = \lceil k / \lfloor \mathrm{FLINT\_BITS}/b \rfloor \rceil$;
• multiprecision, $b$ a multiple of FLINT_BITS: each field takes $b/\mathrm{FLINT\_BITS}$ words and $N = k b / \mathrm{FLINT\_BITS}$.

In both cases the fields must satisfy $f_j < 2^{b-1}$: the top bit of each field is a guard bit, which is zero in every stored monomial. When $b$ does not divide FLINT_BITS the words are not literally the digits of $\Phi(e)$, because of the unused bits, but all statements below hold word by word, and we keep writing $\Phi$ for the packed form.

The width is chosen by mpoly_fix_bits, which rounds a required number of bits up to the largest $b$ with the same number of words: three variables in lex order with 64-bit words give $b = 21$, two give $b = 32$. A polynomial starts with $b = 8$ (MPOLY_MIN_BITS), and an operation whose result needs larger fields repacks with a larger width. All monomials of a polynomial use the same width, recorded in its bits field.

Comparison#

Let cmpmask be zero for ORD_LEX and ORD_DEGLEX, and for ORD_DEGREVLEX the pattern with all bits of fields $0, \ldots, n-1$ set and those of field $n$ clear (mpoly_get_cmpmask).

So comparing monomials costs at most $N$ word comparisons (mpoly_monomial_cmp), and a single one in the frequent case $N = 1$.

The layout relies only on four properties: multiplying monomials adds fields, comparing them compares the packed words after an exclusive or with a mask, every exponent is one of the fields, and all fields are nonnegative.

Multiplication and overflow#

Since every field, including the degree field, is additive, $\Phi(a + c) = \Phi(a) + \Phi(c)$ as long as no field overflows, so monomials are multiplied by adding words (mpoly_monomial_add). The guard bits make overflow detectable.

In multiprecision the guard bit is the top bit of the most significant word of each field (mpoly_monomial_overflows_mp). Algorithms usually avoid testing: before multiplying $B$ by $C$ they compute the maximum of each field over $B$ and over $C$ (mpoly_max_fields_fmpz), add them and choose $b$ large enough for the sum, so that no product of monomials can overflow. Overflow tests are needed where results are not known in advance, as in exact division and in powering.

Divisibility, halving, minima and maxima#

mpoly_monomial_divides performs this test (word by word, so a borrow never crosses a word boundary unnoticed), and the exact division code uses it to detect a leading monomial of the dividend that is not divisible by that of the divisor. The same trick gives, for square roots (mpoly_monomial_halves): $x^a$ is a square if and only if the lowest bit of each word is zero and $(\Phi(a) \gg 1) \mathbin{\&} M = 0$, since shifting a word right by one moves the lowest bit of each field into the guard bit of the field below. Field-wise minima and maxima are computed without unpacking: in $s = M + \Phi(a) - \Phi(c)$ each field equals $2^{b-1} + a_j - c_j \in [1, 2^b)$, so there are no borrows and the guard bit of field $j$ is set exactly when $a_j \ge c_j$; spreading it over the field gives a mask selecting $a_j$ or $c_j$ (mpoly_monomial_min1, mpoly_monomial_max).

Other representations#

The distributed representation suits arithmetic, but some algorithms want a recursive or dense view, to which FLINT converts temporarily:

• Recursive sparse: nmod_mpolyu_t and its relatives are sparse polynomials in one or two main variables whose coefficients are sparse polynomials in the others, and nmod_mpolyn_t is a sparse polynomial whose coefficients are dense univariate polynomials in the last variable. Brown's GCD algorithm works with the latter, Zippel's with the former (Greatest common divisors).
• Univariate with multivariate coefficients: fmpz_mpoly_univar_t and its relatives, public types used for resultants, discriminants and contents.
• Dense: fmpz_mpolyd_t stores all coefficients in a box of given degree bounds and is used by fmpz_mpoly_mul_dense; the array methods of multiplication and division use temporary dense arrays of coefficients indexed by the exponents (Sparse and dense multiplication).
• Dense bivariate: fmpz_bpoly_t, n_bpoly_t and their relatives, arrays of univariate polynomials, used in bivariate factorisation and resultants.

Deflation, compression and repacking#

Many algorithms first simplify the exponents. Write $s_i$ for the minimal exponent of $x_i$ in $A$ and $t_i$ for the GCD of the differences $e_i - s_i$ over the terms. Then $A = x^s \, \iota_t(A')$ where $\iota_t$ is the substitution $x_i \mapsto x_i^{t_i}$ and $A'$ is the deflated polynomial, with exponents $(e_i - s_i)/t_i$ (mpoly_gcd_info_stride, fmpz_mpoly_deflation, fmpz_mpoly_deflate, fmpz_mpoly_inflate).

Thus the GCD of $A = x^{s} \iota(A')$ and $B = x^{s'} \iota(B')$ with common strides $t$ is $x^{\min(s, s')} \iota(\gcd(A', B'))$, since $\iota(A')$ and $\iota(B')$ are not divisible by any variable. The GCD code uses this to reduce degrees. The factoring code goes further and applies an invertible integer affine map to the exponents (compression, mpoly_compression_set), described in Compression.

When exponents need more than one word per field, most algorithms first try to repack them into one word (fmpz_mpoly_repack_bits), which succeeds whenever all fields are below $2^{\mathrm{FLINT\_BITS}-1}$; the GCD code also deflates. Algorithms that only exist for single-word fields (array multiplication, dense methods, GCD and factoring internals) decline or fail otherwise.

Use in FLINT#

• Contexts and layout: mpoly_ctx_init (field order, nfields), mpoly_words_per_exp, mpoly_fix_bits, mpoly_get_cmpmask, mpoly_overflow_mask_sp; packing and unpacking with mpoly_pack_vec_ui, mpoly_set_monomial_ui, mpoly_get_monomial_ui and their fmpz versions; repacking with mpoly_repack_monomials.
• Monomial operations (Lemma 1, Lemma 2, Lemma 3): mpoly_monomial_cmp, mpoly_monomial_add, mpoly_monomial_overflows, mpoly_monomial_divides, mpoly_monomial_halves, mpoly_monomial_max, mpoly_monomial_min1 and their _mp versions.
• Deflation (Lemma 4): mpoly_gcd_info_stride, mpoly_to_mpolyl_perm_deflate, mpoly_from_mpolyl_perm_inflate, fmpz_mpoly_deflation, fmpz_mpoly_deflate, fmpz_mpoly_inflate.

@misc{flint-doc-algorithms-multivariate-representation,
  author       = {{The FLINT team}},
  title        = {{Monomial orderings and exponent packing}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/algorithms/multivariate/representation.html},
  note         = {Generated 2026-11-02}
}