Fix generic integer alignment

[orebas]

Fixes align_up and align_down for non-power-of-two alignments.

The previous bit-mask implementation only works when the alignment is a power of two, but get_next_batch calls align_up(new_batch, composite * tasks). With non-power-of-two task counts, this can produce a batch size that is not divisible by composite * tasks, which can under-allocate task-local primes in the QQ learn-and-apply modular path.

Changes:

• Use cld(x, n) * n and fld(x, n) * n for generic integer alignment.
• Strengthen the learn-and-apply batch invariant to check divisibility by composite * tasks.
• Add regression coverage for non-power-of-two alignments and get_next_batch across tasks=1:16.

Tested with:

• env JULIA_NUM_THREADS=5 julia --project=. bioh_gb_input.jl
  • Groebner.groebner(polys) completed successfully.
  • The script still fails afterward at quotient_basis(gb) because the ideal is not zero-dimensional.
• env JULIA_NUM_THREADS=5 julia --project=. -e 'using Pkg; Pkg.test(Groebner)'
  • 130033 / 130033 tests passed.

[orebas]

Full disclosure: the above PR, including the text, was authored by Codex. It was in response to a crash related to something I was trying, related to the biohydrogenation system. Below is an MWE for the crash.

Obviously, the coefficients are quite large, and you should know that the bug does not surface with "normal" sized coefficients.

I (human Oren) do not understand the bug or the fix.

# Auto-generated by SIAN.jl debug dump.
# Reproduces the Groebner.jl crash on a known polynomial system.
using Nemo
using Groebner
R, _gens = Nemo.polynomial_ring(Nemo.QQ, ["x5_6", "x4_5", "x6_5", "x5_5", "x4_4", "x6_4", "x5_4", "x4_3", "x6_3", "x5_3", "x4_2", "x6_2", "x5_2", "x4_1", "x6_1", "x5_1", "x4_0", "x6_0", "x5_0", "z_aux", "x7_0", "k7_0", "k8_0", "k5_0", "k6_0", "k10_0", "k9_0"], internal_ordering=:degrevlex)
x5_6 = _gens[1]
x4_5 = _gens[2]
x6_5 = _gens[3]
x5_5 = _gens[4]
x4_4 = _gens[5]
x6_4 = _gens[6]
x5_4 = _gens[7]
x4_3 = _gens[8]
x6_3 = _gens[9]
x5_3 = _gens[10]
x4_2 = _gens[11]
x6_2 = _gens[12]
x5_2 = _gens[13]
x4_1 = _gens[14]
x6_1 = _gens[15]
x5_1 = _gens[16]
x4_0 = _gens[17]
x6_0 = _gens[18]
x5_0 = _gens[19]
z_aux = _gens[20]
x7_0 = _gens[21]
k7_0 = _gens[22]
k8_0 = _gens[23]
k5_0 = _gens[24]
k6_0 = _gens[25]
k10_0 = _gens[26]
k9_0 = _gens[27]
polys = [
    -1//2*x5_0 + 4880317582101//2,
    x5_1*x4_0*x6_0 + x5_1*x4_0*x5_0 + 6//5*x4_0*x5_0*k7_0 + 4*x5_1*x4_0*k8_0 - 2*x4_0*x6_0*k5_0 - 2*x4_0*x5_0*k5_0 - 8*x4_0*k8_0*k5_0 + 1//2*x5_1*x6_0*k6_0 + 1//2*x5_1*x5_0*k6_0 + 3//5*x5_0*k7_0*k6_0 + 2*x5_1*k8_0*k6_0,
    -8*x4_0 + 48209756664984,
    x4_1*x4_0 + 1//8*x4_0*k5_0 + 1//2*x4_1*k6_0,
    -1//2*x5_1 + 409269058848616778870403845270905065864//139343436991344010402018855,
    x6_1*x5_1*x4_0 + x5_1^2*x4_0 + x4_1*x5_1*x6_0 + x5_2*x4_0*x6_0 + x4_1*x5_1*x5_0 + x5_2*x4_0*x5_0 + 6//5*x5_1*x4_0*k7_0 + 6//5*x4_1*x5_0*k7_0 + 4*x4_1*x5_1*k8_0 + 4*x5_2*x4_0*k8_0 - 2*x6_1*x4_0*k5_0 - 2*x5_1*x4_0*k5_0 - 2*x4_1*x6_0*k5_0 - 2*x4_1*x5_0*k5_0 - 8*x4_1*k8_0*k5_0 + 1//2*x6_1*x5_1*k6_0 + 1//2*x5_1^2*k6_0 + 1//2*x5_2*x6_0*k6_0 + 1//2*x5_2*x5_0*k6_0 + 3//5*x5_1*k7_0*k6_0 + 2*x5_2*k8_0*k6_0,
    -1//50*x6_0^3*k9_0 - 1//50*x6_0^2*x5_0*k9_0 - 2//25*x6_0^2*k8_0*k9_0 + 2//5*x6_0^2*k10_0*k9_0 + 2//5*x6_0*x5_0*k10_0*k9_0 + 8//5*x6_0*k8_0*k10_0*k9_0 + x6_1*x6_0*k10_0 + x6_1*x5_0*k10_0 - 6//5*x5_0*k7_0*k10_0 + 4*x6_1*k8_0*k10_0,
    -8*x4_1 - 9571718038817124789062058//3153548518895,
    x4_1^2 + x4_2*x4_0 + 1//8*x4_1*k5_0 + 1//2*x4_2*k6_0,
    -1//2*x5_2 - 14109439051198855840348304664214689218404962144636994102589272027671167182732218298215825325770424491//296644081162376242495579934551720237834765136332628598789242776870903991988500,
    2*x4_1*x6_1*x5_1 + 2*x4_1*x5_1^2 + 2*x5_2*x6_1*x4_0 + x6_2*x5_1*x4_0 + 3*x5_2*x5_1*x4_0 + 2*x5_2*x4_1*x6_0 + x4_2*x5_1*x6_0 + x5_3*x4_0*x6_0 + 2*x5_2*x4_1*x5_0 + x4_2*x5_1*x5_0 + x5_3*x4_0*x5_0 + 12//5*x4_1*x5_1*k7_0 + 6//5*x5_2*x4_0*k7_0 + 6//5*x4_2*x5_0*k7_0 + 8*x5_2*x4_1*k8_0 + 4*x4_2*x5_1*k8_0 + 4*x5_3*x4_0*k8_0 - 4*x4_1*x6_1*k5_0 - 4*x4_1*x5_1*k5_0 - 2*x6_2*x4_0*k5_0 - 2*x5_2*x4_0*k5_0 - 2*x4_2*x6_0*k5_0 - 2*x4_2*x5_0*k5_0 - 8*x4_2*k8_0*k5_0 + x5_2*x6_1*k6_0 + 1//2*x6_2*x5_1*k6_0 + 3//2*x5_2*x5_1*k6_0 + 1//2*x5_3*x6_0*k6_0 + 1//2*x5_3*x5_0*k6_0 + 3//5*x5_2*k7_0*k6_0 + 2*x5_3*k8_0*k6_0,
    -3//50*x6_1*x6_0^2*k9_0 - 1//50*x5_1*x6_0^2*k9_0 - 1//25*x6_1*x6_0*x5_0*k9_0 - 4//25*x6_1*x6_0*k8_0*k9_0 + 4//5*x6_1*x6_0*k10_0*k9_0 + 2//5*x5_1*x6_0*k10_0*k9_0 + 2//5*x6_1*x5_0*k10_0*k9_0 + 8//5*x6_1*k8_0*k10_0*k9_0 + x6_1^2*k10_0 + x6_1*x5_1*k10_0 + x6_2*x6_0*k10_0 + x6_2*x5_0*k10_0 - 6//5*x5_1*k7_0*k10_0 + 4*x6_2*k8_0*k10_0,
    -8*x4_2 + 1067558617771569924924858188627476568797421941089//125446498300643216830676606270199069500,
    3*x4_2*x4_1 + x4_3*x4_0 + 1//8*x4_2*k5_0 + 1//2*x4_3*k6_0,
    -1//2*x5_3 + 76860065858484697565538594401685742970632511075807575133062105342768323475594626255438481081350600478195426200413915431990750392614514349630954149451187550061968291//1263033448703247727210485540573994528137545331491467517231722724931844496613846599740786088673792850727889057472356526317929900000,
    6*x5_2*x4_1*x6_1 + 3*x6_2*x4_1*x5_1 + 9*x5_2*x4_1*x5_1 + 3*x4_2*x6_1*x5_1 + 3*x4_2*x5_1^2 + 3*x6_2*x5_2*x4_0 + 3*x5_2^2*x4_0 + 3*x5_3*x6_1*x4_0 + x6_3*x5_1*x4_0 + 4*x5_3*x5_1*x4_0 + 3*x4_2*x5_2*x6_0 + 3*x5_3*x4_1*x6_0 + x4_3*x5_1*x6_0 + x5_4*x4_0*x6_0 + 3*x4_2*x5_2*x5_0 + 3*x5_3*x4_1*x5_0 + x4_3*x5_1*x5_0 + x5_4*x4_0*x5_0 + 18//5*x5_2*x4_1*k7_0 + 18//5*x4_2*x5_1*k7_0 + 6//5*x5_3*x4_0*k7_0 + 6//5*x4_3*x5_0*k7_0 + 12*x4_2*x5_2*k8_0 + 12*x5_3*x4_1*k8_0 + 4*x4_3*x5_1*k8_0 + 4*x5_4*x4_0*k8_0 - 6*x6_2*x4_1*k5_0 - 6*x5_2*x4_1*k5_0 - 6*x4_2*x6_1*k5_0 - 6*x4_2*x5_1*k5_0 - 2*x6_3*x4_0*k5_0 - 2*x5_3*x4_0*k5_0 - 2*x4_3*x6_0*k5_0 - 2*x4_3*x5_0*k5_0 - 8*x4_3*k8_0*k5_0 + 3//2*x6_2*x5_2*k6_0 + 3//2*x5_2^2*k6_0 + 3//2*x5_3*x6_1*k6_0 + 1//2*x6_3*x5_1*k6_0 + 2*x5_3*x5_1*k6_0 + 1//2*x5_4*x6_0*k6_0 + 1//2*x5_4*x5_0*k6_0 + 3//5*x5_3*k7_0*k6_0 + 2*x5_4*k8_0*k6_0,
    -3//25*x6_1^2*x6_0*k9_0 - 2//25*x6_1*x5_1*x6_0*k9_0 - 3//50*x6_2*x6_0^2*k9_0 - 1//50*x5_2*x6_0^2*k9_0 - 1//25*x6_1^2*x5_0*k9_0 - 1//25*x6_2*x6_0*x5_0*k9_0 - 4//25*x6_1^2*k8_0*k9_0 - 4//25*x6_2*x6_0*k8_0*k9_0 + 4//5*x6_1^2*k10_0*k9_0 + 4//5*x6_1*x5_1*k10_0*k9_0 + 4//5*x6_2*x6_0*k10_0*k9_0 + 2//5*x5_2*x6_0*k10_0*k9_0 + 2//5*x6_2*x5_0*k10_0*k9_0 + 8//5*x6_2*k8_0*k10_0*k9_0 + 3*x6_2*x6_1*k10_0 + x5_2*x6_1*k10_0 + 2*x6_2*x5_1*k10_0 + x6_3*x6_0*k10_0 + x6_3*x5_0*k10_0 - 6//5*x5_2*k7_0*k10_0 + 4*x6_3*k8_0*k10_0,
    -1//2*x5_4 + 458517882177903101734911209679018539720563875942963292559534235940453039898464983758817502664647658986541476729730857910488865875026283108160501417089527218191613388035239018644457403026774774508866737592147738230259213229233659//5377668370433502512763376950632568048908562294677487800726728501765858310627819949544983880681369093335742371482553514001380843845033733262655011282018075023618858888820922260000000,
    12*x6_2*x5_2*x4_1 + 12*x5_2^2*x4_1 + 12*x4_2*x5_2*x6_1 + 12*x5_3*x4_1*x6_1 + 6*x4_2*x6_2*x5_1 + 18*x4_2*x5_2*x5_1 + 4*x6_3*x4_1*x5_1 + 16*x5_3*x4_1*x5_1 + 4*x4_3*x6_1*x5_1 + 4*x4_3*x5_1^2 + 6*x5_3*x6_2*x4_0 + 4*x6_3*x5_2*x4_0 + 10*x5_3*x5_2*x4_0 + 4*x5_4*x6_1*x4_0 + x6_4*x5_1*x4_0 + 5*x5_4*x5_1*x4_0 + 6*x5_3*x4_2*x6_0 + 4*x4_3*x5_2*x6_0 + 4*x5_4*x4_1*x6_0 + x4_4*x5_1*x6_0 + x5_5*x4_0*x6_0 + 6*x5_3*x4_2*x5_0 + 4*x4_3*x5_2*x5_0 + 4*x5_4*x4_1*x5_0 + x4_4*x5_1*x5_0 + x5_5*x4_0*x5_0 + 36//5*x4_2*x5_2*k7_0 + 24//5*x5_3*x4_1*k7_0 + 24//5*x4_3*x5_1*k7_0 + 6//5*x5_4*x4_0*k7_0 + 6//5*x4_4*x5_0*k7_0 + 24*x5_3*x4_2*k8_0 + 16*x4_3*x5_2*k8_0 + 16*x5_4*x4_1*k8_0 + 4*x4_4*x5_1*k8_0 + 4*x5_5*x4_0*k8_0 - 12*x4_2*x6_2*k5_0 - 12*x4_2*x5_2*k5_0 - 8*x6_3*x4_1*k5_0 - 8*x5_3*x4_1*k5_0 - 8*x4_3*x6_1*k5_0 - 8*x4_3*x5_1*k5_0 - 2*x6_4*x4_0*k5_0 - 2*x5_4*x4_0*k5_0 - 2*x4_4*x6_0*k5_0 - 2*x4_4*x5_0*k5_0 - 8*x4_4*k8_0*k5_0 + 3*x5_3*x6_2*k6_0 + 2*x6_3*x5_2*k6_0 + 5*x5_3*x5_2*k6_0 + 2*x5_4*x6_1*k6_0 + 1//2*x6_4*x5_1*k6_0 + 5//2*x5_4*x5_1*k6_0 + 1//2*x5_5*x6_0*k6_0 + 1//2*x5_5*x5_0*k6_0 + 3//5*x5_4*k7_0*k6_0 + 2*x5_5*k8_0*k6_0,
    -3//25*x6_1^3*k9_0 - 3//25*x6_1^2*x5_1*k9_0 - 9//25*x6_2*x6_1*x6_0*k9_0 - 3//25*x5_2*x6_1*x6_0*k9_0 - 3//25*x6_2*x5_1*x6_0*k9_0 - 3//50*x6_3*x6_0^2*k9_0 - 1//50*x5_3*x6_0^2*k9_0 - 3//25*x6_2*x6_1*x5_0*k9_0 - 1//25*x6_3*x6_0*x5_0*k9_0 - 12//25*x6_2*x6_1*k8_0*k9_0 - 4//25*x6_3*x6_0*k8_0*k9_0 + 12//5*x6_2*x6_1*k10_0*k9_0 + 6//5*x5_2*x6_1*k10_0*k9_0 + 6//5*x6_2*x5_1*k10_0*k9_0 + 4//5*x6_3*x6_0*k10_0*k9_0 + 2//5*x5_3*x6_0*k10_0*k9_0 + 2//5*x6_3*x5_0*k10_0*k9_0 + 8//5*x6_3*k8_0*k10_0*k9_0 + 3*x6_2^2*k10_0 + 3*x6_2*x5_2*k10_0 + 4*x6_3*x6_1*k10_0 + x5_3*x6_1*k10_0 + 3*x6_3*x5_1*k10_0 + x6_4*x6_0*k10_0 + x6_4*x5_0*k10_0 - 6//5*x5_3*k7_0*k10_0 + 4*x6_4*k8_0*k10_0,
    3*x4_2^2 + 4*x4_3*x4_1 + x4_4*x4_0 + 1//8*x4_3*k5_0 + 1//2*x4_4*k6_0,
    -1//2*x5_5 - 16833291222626600472603619967528380140452113264737492134924964543344289615746421901717382613649373770123335965489900463600074811383984685379818086376751427214884915315245982868073003974938292733436976933545778056421728289828557475442243741312430873962527919992839884125636830777696160934581303//22896715151963940317904916950005900142918955616837872456357564722884774008732038288190372338556653439703735877099500023409448859630678957778713155628103132343460897190557158489427552874663266340017274281333688573940905998124000000000,
    30*x4_2*x6_2*x5_2 + 30*x4_2*x5_2^2 + 30*x5_3*x6_2*x4_1 + 20*x6_3*x5_2*x4_1 + 50*x5_3*x5_2*x4_1 + 30*x5_3*x4_2*x6_1 + 20*x4_3*x5_2*x6_1 + 20*x5_4*x4_1*x6_1 + 10*x6_3*x4_2*x5_1 + 40*x5_3*x4_2*x5_1 + 10*x4_3*x6_2*x5_1 + 30*x4_3*x5_2*x5_1 + 5*x6_4*x4_1*x5_1 + 25*x5_4*x4_1*x5_1 + 5*x4_4*x6_1*x5_1 + 5*x4_4*x5_1^2 + 10*x6_3*x5_3*x4_0 + 10*x5_3^2*x4_0 + 10*x5_4*x6_2*x4_0 + 5*x6_4*x5_2*x4_0 + 15*x5_4*x5_2*x4_0 + 5*x5_5*x6_1*x4_0 + x6_5*x5_1*x4_0 + 6*x5_5*x5_1*x4_0 + 10*x4_3*x5_3*x6_0 + 10*x5_4*x4_2*x6_0 + 5*x4_4*x5_2*x6_0 + 5*x5_5*x4_1*x6_0 + x4_5*x5_1*x6_0 + x5_6*x4_0*x6_0 + 10*x4_3*x5_3*x5_0 + 10*x5_4*x4_2*x5_0 + 5*x4_4*x5_2*x5_0 + 5*x5_5*x4_1*x5_0 + x4_5*x5_1*x5_0 + x5_6*x4_0*x5_0 + 12*x5_3*x4_2*k7_0 + 12*x4_3*x5_2*k7_0 + 6*x5_4*x4_1*k7_0 + 6*x4_4*x5_1*k7_0 + 6//5*x5_5*x4_0*k7_0 + 6//5*x4_5*x5_0*k7_0 + 40*x4_3*x5_3*k8_0 + 40*x5_4*x4_2*k8_0 + 20*x4_4*x5_2*k8_0 + 20*x5_5*x4_1*k8_0 + 4*x4_5*x5_1*k8_0 + 4*x5_6*x4_0*k8_0 - 20*x6_3*x4_2*k5_0 - 20*x5_3*x4_2*k5_0 - 20*x4_3*x6_2*k5_0 - 20*x4_3*x5_2*k5_0 - 10*x6_4*x4_1*k5_0 - 10*x5_4*x4_1*k5_0 - 10*x4_4*x6_1*k5_0 - 10*x4_4*x5_1*k5_0 - 2*x6_5*x4_0*k5_0 - 2*x5_5*x4_0*k5_0 - 2*x4_5*x6_0*k5_0 - 2*x4_5*x5_0*k5_0 - 8*x4_5*k8_0*k5_0 + 5*x6_3*x5_3*k6_0 + 5*x5_3^2*k6_0 + 5*x5_4*x6_2*k6_0 + 5//2*x6_4*x5_2*k6_0 + 15//2*x5_4*x5_2*k6_0 + 5//2*x5_5*x6_1*k6_0 + 1//2*x6_5*x5_1*k6_0 + 3*x5_5*x5_1*k6_0 + 1//2*x5_6*x6_0*k6_0 + 1//2*x5_6*x5_0*k6_0 + 3//5*x5_5*k7_0*k6_0 + 2*x5_6*k8_0*k6_0,
    -18//25*x6_2*x6_1^2*k9_0 - 6//25*x5_2*x6_1^2*k9_0 - 12//25*x6_2*x6_1*x5_1*k9_0 - 9//25*x6_2^2*x6_0*k9_0 - 6//25*x6_2*x5_2*x6_0*k9_0 - 12//25*x6_3*x6_1*x6_0*k9_0 - 4//25*x5_3*x6_1*x6_0*k9_0 - 4//25*x6_3*x5_1*x6_0*k9_0 - 3//50*x6_4*x6_0^2*k9_0 - 1//50*x5_4*x6_0^2*k9_0 - 3//25*x6_2^2*x5_0*k9_0 - 4//25*x6_3*x6_1*x5_0*k9_0 - 1//25*x6_4*x6_0*x5_0*k9_0 - 12//25*x6_2^2*k8_0*k9_0 - 16//25*x6_3*x6_1*k8_0*k9_0 - 4//25*x6_4*x6_0*k8_0*k9_0 + 12//5*x6_2^2*k10_0*k9_0 + 12//5*x6_2*x5_2*k10_0*k9_0 + 16//5*x6_3*x6_1*k10_0*k9_0 + 8//5*x5_3*x6_1*k10_0*k9_0 + 8//5*x6_3*x5_1*k10_0*k9_0 + 4//5*x6_4*x6_0*k10_0*k9_0 + 2//5*x5_4*x6_0*k10_0*k9_0 + 2//5*x6_4*x5_0*k10_0*k9_0 + 8//5*x6_4*k8_0*k10_0*k9_0 + 10*x6_3*x6_2*k10_0 + 4*x5_3*x6_2*k10_0 + 6*x6_3*x5_2*k10_0 + 5*x6_4*x6_1*k10_0 + x5_4*x6_1*k10_0 + 4*x6_4*x5_1*k10_0 + x6_5*x6_0*k10_0 + x6_5*x5_0*k10_0 - 6//5*x5_4*k7_0*k10_0 + 4*x6_5*k8_0*k10_0,
    10*x4_3*x4_2 + 5*x4_4*x4_1 + x4_5*x4_0 + 1//8*x4_4*k5_0 + 1//2*x4_5*k6_0,
    -1//2*x5_6 + 2230503213357701357569440907020643973817630698962549348350662729250655449889457673354642280874096489436871587946168655140601401023983404657196226995727560762555964180378041950055367740709808674027261529388384365684009605417002804896233524786236476486842524943013818922830610735730276765570285347120647310652522598122337977161210397249911331395200124487341//3899530641439785758106519031248013429963970214268438235464842731697783528278382630958690014928873154952997481236688863273797382818503720654802833380324793230775125296123999524196161696690367099759313565375256537591412364714890538418076245936124157609372547934041419239011104000000000,
    -8*x4_3 + 9980234898521577864742474945154512255354536124394581060218071638161101613//9980391195254194232908371166727274297961010400800055115489900000,
    -8*x4_4 + 27529340855090325099548923027554684493195513387169307058417215755254308156622223797175741718380427//158805881008472455249978402662551270575313740548855540004693911882975830493225560236000000,
    -8*x4_5 + 499556367342470625025569534582373295139636271001372606136834196241995586886722399915140375746394973158165846793893016126211//12634428525641971425260015905982407389975487650780450039277308668489330286084916530481745199901284447967695200000000,
    x4_0*x6_0*z_aux*k10_0 + x4_0*x5_0*z_aux*k10_0 + 4*x4_0*z_aux*k8_0*k10_0 + 1//2*x6_0*z_aux*k6_0*k10_0 + 1//2*x5_0*z_aux*k6_0*k10_0 + 2*z_aux*k8_0*k6_0*k10_0 - 1,
]

println("Versions: Julia ", VERSION, " Nemo ", pkgversion(Nemo), " Groebner ", pkgversion(Groebner))
println("System: ", length(polys), " polys, ", length(_gens), " vars")
println("Calling Groebner.groebner...")
gb = Groebner.groebner(polys)
println("  ✓ Got GB with ", length(gb), " elements")
println("  quotient_basis dim = ", length(Groebner.quotient_basis(gb)))