Buchberger naive for fmpz_mod_mpoly #2445

andrii302 · 2025-10-29T08:49:36Z

Added a fmpz_mod_mpoly version of Buchberger algorithm used in fmpz_mod_mpoly.

albinahlback · 2025-10-29T09:15:26Z

src/fmpz_mod_mpoly.h

+void _fmpz_mod_mpoly_vec_fmpz_mod_mpoly_to_fmpz_mpoly(fmpz_mpoly_vec_t ret, fmpz_mod_mpoly_vec_t src, fmpz_mod_mpoly_ctx_t ctx);
+void _fmpz_mod_mpoly_vec_fmpz_mpoly_to_fmpz_mod_mpoly(fmpz_mod_mpoly_vec_t ret, fmpz_mpoly_vec_t src, fmpz_mod_mpoly_ctx_t ctx);


Where are these defined? I can't seem to find it.

albinahlback · 2025-10-29T09:16:34Z

src/fmpz_mod_mpoly/buchberger.c

Remove this file if it is unused

albinahlback · 2025-10-29T09:18:06Z

src/fmpz_mod_mpoly.h

 #endif

 #include "fmpz.h"
+#include "fmpz_mpoly.h"


I don't think this inclusion is necessary.

fredrik-johansson · 2025-10-29T09:58:28Z

src/fmpz_mod_mpoly/buchberger_naive.c

+    if (fmpz_mod_mpoly_length(poly, ctx) > poly_len_limit)
+        return 0;
+
+    bits = _fmpz_vec_max_bits(poly->coeffs, poly->length);


This is no longer needed.

fredrik-johansson · 2025-10-29T09:59:45Z

doc/source/fmpz_mod_mpoly.rst

+-------------------------------------------------------------------------------
+
+The following methods deal with ideals in `\mathbb{Z}/n\mathbb{Z}[x_1, \dots, x_m]`.
+We use primitive integer polynomials as normalised generators


The normalised generators will be monic, not primitive integer polynomials.

fredrik-johansson · 2025-10-29T10:01:15Z

src/fmpz_mod_mpoly/spoly.c

+    for (i = 0; i < n; i++)
+        exp[i] = FLINT_MAX(expf[i], expg[i]);
+
+    fmpz_mod_inv(c, f->coeffs, ctx->ffinfo);


You can check if the leading coefficients are 1, and then you don't need to invert them and multiply by them below.

fredrik-johansson · 2025-10-29T10:02:19Z

src/fmpz_mod_mpoly/vec_set_monic_unique.c

+#include "fmpz_mod_mpoly.h"
+
+void
+fmpz_mod_mpoly_vec_set_monic_unique(fmpz_mod_mpoly_vec_t G, const fmpz_mod_mpoly_vec_t F, const fmpz_mod_mpoly_ctx_t ctx)


File should be renamed to match the function name.

fredrik-johansson · 2025-10-29T10:19:43Z

src/fmpz_mod_mpoly/test/t-buchberger_naive.c

+        fmpz_mod_mpoly_vec_init(H, 0, ctx);
+
+
+        // flint_printf("iter %ld   %ld  %d    %d \n\n", iter, nvars, ctx->minfo->ord, m);


It's just a comment, but you can change the format string to iter %wd %wd %d %{fmpz}

fredrik-johansson · 2025-10-29T10:57:48Z

There is maybe some bug in the algorithm.

I tried to get some bigger examples by setting

        if (nvars == 4)
            fmpz_mod_mpoly_vec_randtest_not_zero(F, state, 1 + n_randint(state, 3), 1 + n_randint(state, 3), 1 + n_randint(state, 3), ctx);
        else if (nvars == 3)
            fmpz_mod_mpoly_vec_randtest_not_zero(F, state, 1 + n_randint(state, 4), 1 + n_randint(state, 4), 1 + n_randint(state, 4), ctx);
        else
            fmpz_mod_mpoly_vec_randtest_not_zero(F, state, 1 + n_randint(state, 6), 1 + n_randint(state, 6), 1 + n_randint(state, 5), ctx);

On iteration 224 we get

iter 224   3  0    4722366482869611659327 

F = [4491077222722822274764*x1^3*x2^2*x3^3+1142989093829542840957*x1*x2*x3+161135877041952360283*x2^2*x3^2+1963195568815792903903*x3^2, 473518656873042772582*x1^3*x2^2*x3^2+538964973504606229287*x1^3*x3^3+2996428526938692536564*x1^2*x2^2*x3^2+1389613235437080610847*x1*x3]

Sage tells me

sage: R.<x1,x2,x3> = PolynomialRing(GF(4722366482869611659327))
sage: F = [4491077222722822274764*x1^3*x2^2*x3^3+1142989093829542840957*x1*x2*x3+161135877041952360283*x2^2*x3^2+1963195568815792903903*x3^2, 473518656873042772582*x1^3*x2^2*x3^2+538964973504606229287*x1^3*x3^3+
....: 2996428526938692536564*x1^2*x2^2*x3^2+1389613235437080610847*x1*x3]
sage: Ideal(F).groebner_basis()
[x2^6*x3^3 + 712996795176620819850*x1^3*x2^4*x3 + 2991372772035270063807*x2^6*x3^2 + 2232673190335890371919*x1^3*x2*x3^4 + 4622290847937109124679*x2^4*x3^4 + 2232118258695245099135*x1*x2^2*x3^5 + 1337247136055595098177*x1^2*x2^4*x3 + 3809242543601178032643*x1*x2^5*x3 + 2255749836307100781426*x1^2*x2^3*x3^2 + 3522766432234796391146*x1^3*x2*x3^3 + 674045606485042985327*x1*x2^3*x3^3 + 144410032056534803816*x2^4*x3^3 + 3742288888964036579985*x1^3*x3^4 + 3806698843246029539861*x1^2*x2*x3^4 + 1261854264738708411457*x1*x2^2*x3^4 + 3174422527955615953712*x1^2*x2^3*x3 + 1309130462284157687072*x1^2*x2^2*x3^2 + 511509324830378656330*x1*x2^3*x3^2 + 3843146600048663289891*x2^4*x3^2 + 1426607345279955621580*x1^3*x3^3 + 2931203426717357998872*x1^2*x2*x3^3 + 3765559273451236887495*x1^2*x3^4 + 1800815570161933110697*x2^2*x3^4 + 4362577392138830451491*x1*x3^5 + 1749266203946741954616*x1*x2^3*x3 + 603762769168935094984*x1^2*x2*x3^2 + 802440712834424206579*x1*x2^2*x3^2 + 2312779589572324370120*x2^3*x3^2 + 2862441150645602461203*x1*x2*x3^3 + 2301513304013421389416*x2^2*x3^3 + 3830223109782334087895*x1*x3^4 + 1845529563375065208582*x1*x2^2*x3 + 3916000751915517880153*x1*x2*x3^2 + 4454114546813219238110*x2^2*x3^2 + 1771132791807173937207*x1*x3^3 + 19748604038564336348*x3^4 + 2619902816384677653484*x1*x2*x3 + 1629376223837254477255*x1*x3^2 + 2341798583917537710307*x2*x3^2 + 452050843028990228886*x3^3 + 3302703954367051778374*x1*x3 + 852676813439846411794*x3^2,
 x1^4*x2^3*x3 + 1426516598602069476079*x1^3*x2^3*x3 + 100075634932502534648*x1^4*x2*x3^2 + 1540191522950982634971*x1^4*x3^3 + 2181333662874204585127*x1^2*x2^2*x3^2 + 2254146332524917002871*x1^3*x3^3 + 1821799783765970911452*x1*x2^2*x3 + 2808277818627968062792*x1^2*x3 + 2504251270690830754814*x1*x3,
 x1^4*x3^4 + 1446493953846792653206*x1*x2^2*x3^2 + 2888952530396186513897*x1^2*x2*x3 + 12295073657520105697*x1^2*x3^2 + 991501011636667918886*x2^2*x3^2 + 1777109014141401984424*x1*x2*x3 + 2577407212188895213663*x1*x3^2 + 172392630374498717073*x3^2,
 x1^3*x3^5 + 1134726875404767220728*x1^2*x2^3*x3^2 + 232381224220883413991*x2^4*x3^3 + 4445847190008855697024*x1^3*x3^4 + 4432758092541225535985*x1*x2^2*x3^4 + 1046038837962674726577*x1*x2^3*x3^2 + 2657044711577393369498*x2^4*x3^2 + 1075309877823930528429*x1^2*x2*x3^3 + 2261868349800318839345*x1^2*x3^4 + 1394206333302585142984*x1*x2^3*x3 + 1698811328637074417449*x1*x2^2*x3^2 + 115279397433147445704*x2^2*x3^3 + 74882451101184199473*x1*x3^4 + 1055538577922761368467*x1*x2*x3^2 + 507487343683842385767*x2^2*x3^2 + 12295073657520105697*x1*x3^3 + 3080264025967845729397*x1*x2*x3 + 2697119451786508761151*x1*x3^2 + 432447941508178767999*x3^3 + 2571238725551279090344*x3^2,
 x1^3*x2^2*x3^2 + 1426516598602069476079*x1^2*x2^2*x3^2 + 100075634932502534648*x1^3*x3^3 + 4035834253805165078146*x1*x3,
 x1*x2^4*x3^2 + 4108625839513552360366*x1^2*x2^3*x3 + 3202032657719516531780*x1^2*x2^2*x3^2 + 1426516598602069476079*x2^4*x3^2 + 100075634932502534648*x1*x2^2*x3^3 + 2503195822188664255580*x1*x2^3*x3 + 4048320876384568674000*x1^2*x2*x3^2 + 898560334578367526013*x1*x2^2*x3^2 + 2945259352879771335346*x2^2*x3^2 + 384260550207056090205*x1*x3^3,
 x1^2*x2^2*x3^3 + 4271013918249353011579*x1^3*x3^4 + 2786951067681631733096*x2^2*x3^2 + 1709684415700564492299*x1*x2*x3 + 2045591994075599314921*x1*x3^2 + 4461913745913676329068*x3^2]

But the vector H computed by fmpz_mod_mpoly_buchberger_naive + fmpz_mod_mpoly_vec_autoreduction_groebner here is something much bigger.

I guess one can find a much smaller bad case; I just took the first really big one I saw that looked suspect.

fredrik-johansson · 2025-10-29T11:13:57Z

More manageable example:

3 variables, mod 17

F = [6*x1^2*x2+14*x2*x3^2+6, 3*x1^2*x2^2*x3+4*x1*x2+3*x3^2, 15*x1^2*x3+2*x1*x2^2+4*x3^2]

H = [x3^21+x3^20+11*x3^19+16*x3^18+8*x3^16+x3^15+6*x3^14+16*x3^13+11*x3^12+4*x3^11+7*x3^10+3*x3^9+16*x3^8+8*x3^7+13*x3^6+5*x3^5+8*x3^3+15*x3^2+10, x1+5*x3^20+6*x3^18+16*x3^17+9*x3^16+10*x3^15+6*x3^14+13*x3^13+16*x3^12+10*x3^11+8*x3^10+4*x3^9+6*x3^8+16*x3^7+6*x3^6+12*x3^4+9*x3^2+7*x3+12, x2+16*x3^20+4*x3^19+14*x3^18+6*x3^17+11*x3^16+9*x3^15+12*x3^14+4*x3^13+8*x3^12+15*x3^11+14*x3^10+12*x3^9+14*x3^8+15*x3^7+15*x3^6+3*x3^5+7*x3^4+16*x3^3+9*x3^2+3*x3+10]

Sage says that H is not a Groebner basis.

fredrik-johansson · 2025-10-29T15:03:10Z

Oh wait, I'm stupid. I forgot about term orders! SageMath doesn't use lex ordering by default. The H is indeed a lex Groebner basis. So I think there's no problem.

fredrik-johansson · 2025-10-29T15:29:48Z

Fun fact: I tried

sage: R.<x1,x2,x3> = PolynomialRing(GF(4722366482869611659327), order="lex")
sage: 
....: F = [4491077222722822274764*x1^3*x2^2*x3^3+1142989093829542840957*x1*x2*x3+161135877041952360283*x2^2*x3^2+1963195568815792903903*x3^2, 473518656873042772582*x1^3*x2^2*x3^2+538964973504606229287*x1^3*x3^3+
....: 2996428526938692536564*x1^2*x2^2*x3^2+1389613235437080610847*x1*x3]
....: 
sage: %time G = Ideal(F).groebner_basis()

and it's still going after 20 minutes. FLINT did this one in 14 seconds. Is this Singular being slow, Sage doing something bad, or FLINT being fast?

Edit: Sage (Singular?) was killed by the OS after about 30 minutes.

fredrik-johansson · 2025-10-29T15:31:49Z

src/fmpz_mod_mpoly/spoly.c

+    {
+        fmpz_t c;
+        fmpz_mod_inv(c, f->coeffs, ctx->ffinfo);
+        fmpz_init(c);


The fmpz_init needs to be moved one line up. Same in the block below.

guyomes · 2025-10-29T16:57:53Z

= PolynomialRing(GF(4722366482869611659327), order="lex")

One way that seems to work more efficiently in Sage is to use the transformed_basis function, that computes a lex Gröbner basis using Gröbner walk. It returns whithin a second on my laptop.

R.<x1,x2,x3>  = PolynomialRing(GF(4722366482869611659327), order='lex')
F = [4491077222722822274764*x1^3*x2^2*x3^3+1142989093829542840957*x1*x2*x3+161135877041952360283*x2^2*x3^2+1963195568815792903903*x3^2, 473518656873042772582*x1^3*x2^2*x3^2+538964973504606229287*x1^3*x3^3+ 2996428526938692536564*x1^2*x2^2*x3^2+1389613235437080610847*x1*x3]

S = R.change_ring(order='degrevlex')
G = (S*F).transformed_basis()

Edit: this basis does not seem reduced though. Reducing it took around 9 seconds on my laptop, and can be done as follows.

Gred = (R*G).groebner_basis()

fredrik-johansson · 2025-10-29T17:03:34Z

Output from FLINT: GB.txt

edgarcosta · 2025-10-29T19:08:46Z

Magma converts between orders "automatically", I do not know the details.

> R<x1,x2,x3> := PolynomialRing(GF(4722366482869611659327), 3, "grevlex");
> F := [4491077222722822274764*x1^3*x2^2*x3^3+1142989093829542840957*x1*x2*x3+161135877041952360283*x2^2*x3^2+1963195568815792903903*x3^2, 473518656873042772582*x1^3*x2^2*x3^2+538964973\
504606229287*x1^3*x3^3+2996428526938692536564*x1^2*x2^2*x3^2+1389613235437080610847*x1*x3];
> time GB := GroebnerBasis(ideal<R|F>);
Time: 0.000
> R<x1,x2,x3> := PolynomialRing(GF(4722366482869611659327), 3, "lex");
> F := [4491077222722822274764*x1^3*x2^2*x3^3+1142989093829542840957*x1*x2*x3+161135877041952360283*x2^2*x3^2+1963195568815792903903*x3^2, 473518656873042772582*x1^3*x2^2*x3^2+538964973\
504606229287*x1^3*x3^3+2996428526938692536564*x1^2*x2^2*x3^2+1389613235437080610847*x1*x3];
> time GB := GroebnerBasis(ideal<R|F>);
Time: 0.030
> R<x1,x2,x3> := PolynomialRing(GF(4722366482869611659327), 3, "lex");
> F := [4491077222722822274764*x1^3*x2^2*x3^3+1142989093829542840957*x1*x2*x3+161135877041952360283*x2^2*x3^2+1963195568815792903903*x3^2, 473518656873042772582*x1^3*x2^2*x3^2+538964973\
> 504606229287*x1^3*x3^3+2996428526938692536564*x1^2*x2^2*x3^2+1389613235437080610847*x1*x3];
> time GB := GroebnerBasis(ideal<R|F> : Al:="Direct");
^C
[Interrupted]
Total time: 688.460 seconds, Total memory usage: 1774.78MB

fredrik-johansson · 2025-10-30T13:43:17Z

Naming:

fmpz_mod_mpoly_from_fmpz_mpoly -> fmpz_mod_mpoly_set_fmpz_mpoly
fmpz_mod_mpoly_to_fmpz_mpoly -> fmpz_mod_mpoly_get_fmpz_mpoly

fredrik-johansson · 2025-10-30T15:12:35Z

Can you undo the changes to fmpz_mpoly/test/t-buchberger_naive.c. Other than that, looks good!

andrii302 and others added 7 commits October 28, 2025 16:46

Added naive Buchberger to fmpz_mod_mpoly

bae70ae

Removed the polybits argument

fcfc899

Merge branch 'main' of https://github.com/andrii302/flint

877e13c

Removed unused argument from buchberger

783de7b

Merge branch 'flintlib:main' into main

b27d235

Merge branch 'main' of https://github.com/andrii302/flint

1439b5f

Added documentation

63c70dd

albinahlback reviewed Oct 29, 2025

View reviewed changes

minor fixes

2b0b6db

fredrik-johansson reviewed Oct 29, 2025

View reviewed changes

andrii302 added 2 commits October 29, 2025 11:50

minor changes

13d15f3

Commented out print in test

6d752c9

fredrik-johansson reviewed Oct 29, 2025

View reviewed changes

andrii302 added 4 commits October 30, 2025 13:45

Added to_from_fmpz to fmpz_mod_mpoly

ec10b08

small fix

e5e2d55

Added documentation

f83245c

minor fixes

d9d1d99

Renamed from/to functions

adf12ba

unchanged fmpz_mpoly_buchberger_naive

e442d35

		void _fmpz_mod_mpoly_vec_fmpz_mod_mpoly_to_fmpz_mpoly(fmpz_mpoly_vec_t ret, fmpz_mod_mpoly_vec_t src, fmpz_mod_mpoly_ctx_t ctx);
		void _fmpz_mod_mpoly_vec_fmpz_mpoly_to_fmpz_mod_mpoly(fmpz_mod_mpoly_vec_t ret, fmpz_mpoly_vec_t src, fmpz_mod_mpoly_ctx_t ctx);

		fmpz_mod_mpoly_vec_init(H, 0, ctx);


		// flint_printf("iter %ld %ld %d %d \n\n", iter, nvars, ctx->minfo->ord, m);

Buchberger naive for fmpz_mod_mpoly #2445

Are you sure you want to change the base?

Buchberger naive for fmpz_mod_mpoly #2445

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