sagemathgh-39075: Allow more rings to be used with libsingular & rework signal detection system
    
Somewhat related: sagemath#33319

Anyway, this allows many more rings to be used with libsingular.
Previously it fallback to the expect interface.

It seems weird that both Sage and Singular has different types for
`GF(5)` and `Zmod(5)`, but okay.

### 📝 Checklist

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- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: sagemath#39075
Reported by: user202729
Reviewer(s): Martin Rubey, user202729

Author: Release Manager

src/sage/libs/singular/function.pyx

@@ -97,7 +97,7 @@ from sage.rings.polynomial.multi_polynomial_sequence import PolynomialSequence_g
 from sage.libs.singular.decl cimport *
 from sage.libs.singular.option import opt_ctx
 from sage.libs.singular.polynomial cimport singular_vector_maximal_component
-from sage.libs.singular.singular cimport sa2si, si2sa, si2sa_intvec, si2sa_bigintvec
+from sage.libs.singular.singular cimport sa2si, si2sa, si2sa_intvec, si2sa_bigintvec, start_catch_error, check_error
 from sage.libs.singular.singular import error_messages
 
 from sage.interfaces.singular import get_docstring
@@ -1450,10 +1450,8 @@ EXAMPLES::
 
 cdef inline call_function(SingularFunction self, tuple args, object R, bint signal_handler=True, attributes=None):
     global currRingHdl
-    global errorreported
     global currentVoice
     global myynest
-    global error_messages
 
     cdef ring *si_ring
     if isinstance(R, MPolynomialRing_libsingular):
@@ -1474,29 +1472,28 @@ cdef inline call_function(SingularFunction self, tuple args, object R, bint sign
 
     currentVoice = NULL
     myynest = 0
-    errorreported = 0
-
-    while error_messages:
-        error_messages.pop()
+    start_catch_error()
 
     with opt_ctx: # we are preserving the global options state here
         if signal_handler:
-            sig_on()
-            _res = self.call_handler.handle_call(argument_list, si_ring)
-            sig_off()
+            try:
+                sig_on()
+                _res = self.call_handler.handle_call(argument_list, si_ring)
+                sig_off()
+            finally:
+                s = check_error()
         else:
             _res = self.call_handler.handle_call(argument_list, si_ring)
+            s = check_error()
 
-    if myynest:
-        myynest = 0
+    myynest = 0
 
     if currentVoice:
         currentVoice = NULL
 
-    if errorreported:
-        errorreported = 0
+    if s:
         raise RuntimeError("error in Singular function call %r:\n%s" %
-            (self._name, "\n".join(error_messages)))
+                           (self._name, "\n".join(s)))
 
     res = argument_list.to_python(_res)
 

src/sage/libs/singular/ring.pyx

@@ -19,7 +19,7 @@ AUTHORS:
 from sage.cpython.string cimport str_to_bytes, bytes_to_str
 
 from sage.libs.gmp.types cimport __mpz_struct
-from sage.libs.gmp.mpz cimport mpz_init_set_ui
+from sage.libs.gmp.mpz cimport mpz_init_set
 
 from sage.libs.singular.decl cimport ring, currRing
 from sage.libs.singular.decl cimport rChangeCurrRing, rComplete, rDelete, idInit
@@ -31,6 +31,7 @@ from sage.libs.singular.decl cimport n_coeffType
 from sage.libs.singular.decl cimport rDefault, GFInfo, ZnmInfo, nInitChar, AlgExtInfo, TransExtInfo
 
 
+from sage.rings.integer cimport Integer
 from sage.rings.integer_ring cimport IntegerRing_class
 from sage.rings.integer_ring import ZZ
 import sage.rings.abc
@@ -80,7 +81,7 @@ if bytes_to_str(rSimpleOrdStr(ringorder_ip)) == "rp":
 
 #############################################################################
 cdef ring *singular_ring_new(base_ring, n, names, term_order) except NULL:
-    """
+    r"""
     Create a new Singular ring over the ``base_ring`` in ``n``
     variables with the names ``names`` and the term order
     ``term_order``.
@@ -159,17 +160,136 @@ cdef ring *singular_ring_new(base_ring, n, names, term_order) except NULL:
         sage: R.<x,y,z> = F[]
         sage: from sage.libs.singular.function import singular_function
         sage: sing_print = singular_function('print')
-        sage: sing_print(R)
-        'polynomial ring, over a field, global ordering\n// coefficients: ZZ/7(a, b)\n// number of vars : 3\n//        block   1 : ordering dp\n//                  : names    x y z\n//        block   2 : ordering C'
+        sage: print(sing_print(R))
+        polynomial ring, over a field, global ordering
+        // coefficients: ZZ/7(a, b)
+        // number of vars : 3
+        //        block   1 : ordering dp
+        //                  : names    x y z
+        //        block   2 : ordering C
 
     ::
 
         sage: F = PolynomialRing(QQ, 's,t').fraction_field()
         sage: R.<x,y,z> = F[]
         sage: from sage.libs.singular.function import singular_function
-        sage: sing_print = singular_function('print')
-        sage: sing_print(R)
-        'polynomial ring, over a field, global ordering\n// coefficients: QQ(s, t)\n// number of vars : 3\n//        block   1 : ordering dp\n//                  : names    x y z\n//        block   2 : ordering C'
+        sage: print(sing_print(R))
+        polynomial ring, over a field, global ordering
+        // coefficients: QQ(s, t)
+        // number of vars : 3
+        //        block   1 : ordering dp
+        //                  : names    x y z
+        //        block   2 : ordering C
+
+    Small primes::
+
+        sage: R = PolynomialRing(GF(2), ("a", "b"), implementation="singular"); print(sing_print(R))
+        polynomial ring, over a field, global ordering
+        // coefficients: ZZ/2
+        // number of vars : 2
+        //        block   1 : ordering dp
+        //                  : names    a b
+        //        block   2 : ordering C
+        sage: R = PolynomialRing(GF(3), ("a", "b"), implementation="singular"); print(sing_print(R))
+        polynomial ring, over a field, global ordering
+        // coefficients: ZZ/3
+        // number of vars : 2
+        //        block   1 : ordering dp
+        //                  : names    a b
+        //        block   2 : ordering C
+        sage: R = PolynomialRing(GF(1000000007), ("a", "b"), implementation="singular"); print(sing_print(R))
+        polynomial ring, over a field, global ordering
+        // coefficients: ZZ/1000000007
+        // number of vars : 2
+        //        block   1 : ordering dp
+        //                  : names    a b
+        //        block   2 : ordering C
+
+    When ``Zmod`` is used, use a different Singular type
+    (note that the print is wrong, the field in fact doesn't have zero-divisors)::
+
+        sage: R = PolynomialRing(Zmod(2), ("a", "b"), implementation="singular"); print(sing_print(R))
+        polynomial ring, over a ring (with zero-divisors), global ordering
+        // coefficients: ZZ/(2)
+        // number of vars : 2
+        //        block   1 : ordering dp
+        //                  : names    a b
+        //        block   2 : ordering C
+        sage: R = PolynomialRing(Zmod(3), ("a", "b"), implementation="singular"); print(sing_print(R))
+        polynomial ring, over a ring (with zero-divisors), global ordering
+        // coefficients: ZZ/(3)
+        // number of vars : 2
+        //        block   1 : ordering dp
+        //                  : names    a b
+        //        block   2 : ordering C
+
+    Large prime (note that the print is wrong, the field in fact doesn't have zero-divisors)::
+
+        sage: R = PolynomialRing(GF(2^128+51), ("a", "b"), implementation="singular"); print(sing_print(R))
+        polynomial ring, over a ring (with zero-divisors), global ordering
+        // coefficients: ZZ/bigint(340282366920938463463374607431768211507)
+        // number of vars : 2
+        //        block   1 : ordering dp
+        //                  : names    a b
+        //        block   2 : ordering C
+
+    Finite field with large degree (note that if stack size is too small and the exponent is too large
+    a stack overflow may happen inside libsingular)::
+
+        sage: R = PolynomialRing(GF(2^160), ("a", "b"), implementation="singular"); print(sing_print(R))
+        polynomial ring, over a field, global ordering
+        // coefficients: ZZ/2[z160]/(z160^160+z160^159+z160^155+z160^154+z160^153+z160^152+z160^151+z160^149+z160^148+z160^147+z160^146+z160^145+z160^144+z160^143+z160^141+z160^139+z160^137+z160^131+z160^129+z160^128+z160^127+z160^126+z160^123+z160^122+z160^121+z160^117+z160^116+z160^115+z160^113+z160^111+z160^110+z160^108+z160^106+z160^102+z160^100+z160^99+z160^97+z160^96+z160^95+z160^94+z160^93+z160^92+z160^91+z160^87+z160^86+z160^82+z160^80+z160^79+z160^78+z160^74+z160^73+z160^72+z160^71+z160^70+z160^67+z160^66+z160^65+z160^62+z160^59+z160^58+z160^57+z160^55+z160^54+z160^53+z160^52+z160^51+z160^49+z160^47+z160^44+z160^40+z160^35+z160^32+z160^30+z160^28+z160^27+z160^26+z160^24+z160^23+z160^21+z160^20+z160^18+z160^16+z160^11+z160^10+z160^8+z160^7+1)
+        // number of vars : 2
+        //        block   1 : ordering dp
+        //                  : names    a b
+        //        block   2 : ordering C
+
+    Integer modulo small power of 2::
+
+        sage: R = PolynomialRing(Zmod(2^32), ("a", "b"), implementation="singular"); print(sing_print(R))
+        polynomial ring, over a ring (with zero-divisors), global ordering
+        // coefficients: ZZ/(2^32)
+        // number of vars : 2
+        //        block   1 : ordering dp
+        //                  : names    a b
+        //        block   2 : ordering C
+
+    Integer modulo large power of 2::
+
+        sage: R = PolynomialRing(Zmod(2^1000), ("a", "b"), implementation="singular"); print(sing_print(R))
+        polynomial ring, over a ring (with zero-divisors), global ordering
+        // coefficients: ZZ/(bigint(2)^1000)
+        // number of vars : 2
+        //        block   1 : ordering dp
+        //                  : names    a b
+        //        block   2 : ordering C
+
+    Integer modulo large power of odd prime::
+
+        sage: R = PolynomialRing(Zmod(3^300), ("a", "b"), implementation="singular"); print(sing_print(R))
+        polynomial ring, over a ring (with zero-divisors), global ordering
+        // coefficients: ZZ/(bigint(3)^300)
+        // number of vars : 2
+        //        block   1 : ordering dp
+        //                  : names    a b
+        //        block   2 : ordering C
+
+    Integer modulo non-prime::
+
+        sage: R = PolynomialRing(Zmod(15^20), ("a", "b"), implementation="singular"); print(sing_print(R))
+        polynomial ring, over a ring (with zero-divisors), global ordering
+        // coefficients: ZZ/bigint(332525673007965087890625)
+        // number of vars : 2
+        //        block   1 : ordering dp
+        //                  : names    a b
+        //        block   2 : ordering C
+
+    Non-prime finite field with large characteristic (not supported, see :issue:`33319`)::
+
+        sage: PolynomialRing(GF((2^31+11)^2), ("a", "b"), implementation="singular")
+        Traceback (most recent call last):
+        ...
+        TypeError: characteristic must be <= 2147483647.
     """
     cdef long cexponent
     cdef GFInfo* _param
@@ -182,7 +302,7 @@ cdef ring *singular_ring_new(base_ring, n, names, term_order) except NULL:
     cdef int offset
     cdef int nvars
     cdef int characteristic
-    cdef int modbase
+    cdef Integer ch, modbase
     cdef int ringorder_column_pos
     cdef int ringorder_column_asc
 
@@ -377,39 +497,57 @@ cdef ring *singular_ring_new(base_ring, n, names, term_order) except NULL:
         _cf = nInitChar( n_Z, NULL) # integer coefficient ring
         _ring = rDefault (_cf, nvars, _names, nblcks, _order, _block0, _block1, _wvhdl)
 
-    elif (isinstance(base_ring, FiniteField_generic) and base_ring.is_prime_field()):
-        if base_ring.characteristic() <= 2147483647:
+    elif isinstance(base_ring, sage.rings.abc.IntegerModRing):
+
+        ch = base_ring.characteristic()
+        if ch < 2:
+            raise NotImplementedError(f"polynomials over {base_ring} are not supported in Singular")
+
+        isprime = ch.is_prime()
+
+        if isprime and ch <= 2147483647 and isinstance(base_ring, FiniteField_generic):
+            # don't use this branch for e.g. Zmod(5)
             characteristic = base_ring.characteristic()
+
+            # example for simpler ring creation interface without monomial orderings:
+            #_ring = rDefault(characteristic, nvars, _names)
+
+            _ring = rDefault(characteristic, nvars, _names, nblcks, _order, _block0, _block1, _wvhdl)
+
         else:
-            raise TypeError("Characteristic p must be <= 2147483647.")
+            modbase, cexponent = ch.perfect_power()
 
-        # example for simpler ring creation interface without monomial orderings:
-        #_ring = rDefault(characteristic, nvars, _names)
+            if modbase == 2 and cexponent > 1:
+                _cf = nInitChar(n_Z2m, <void *>cexponent)
 
-        _ring = rDefault(characteristic, nvars, _names, nblcks, _order, _block0, _block1, _wvhdl)
+            elif modbase.is_prime() and cexponent > 1:
+                _info.base = <__mpz_struct *>omAlloc(sizeof(__mpz_struct))
+                mpz_init_set(_info.base, modbase.value)
+                _info.exp = cexponent
+                _cf = nInitChar(n_Znm, <void *>&_info)
+
+            else:
+                _info.base = <__mpz_struct *>omAlloc(sizeof(__mpz_struct))
+                mpz_init_set(_info.base, ch.value)
+                _info.exp = 1
+                _cf = nInitChar(n_Zn, <void *>&_info)
+            _ring = rDefault(_cf, nvars, _names, nblcks, _order, _block0, _block1, _wvhdl)
 
     elif isinstance(base_ring, FiniteField_generic):
-        if base_ring.characteristic() <= 2147483647:
-            characteristic = -base_ring.characteristic() # note the negative characteristic
-        else:
+        assert not base_ring.is_prime_field()  # would have been handled above
+        if base_ring.characteristic() > 2147483647:
             raise TypeError("characteristic must be <= 2147483647.")
 
         # TODO: This is lazy, it should only call Singular stuff not PolynomialRing()
         k = PolynomialRing(base_ring.prime_subfield(),
-            name=base_ring.variable_name(), order='lex', implementation='singular')
+                           name=base_ring.variable_name(), order='lex',
+                           implementation='singular')
         minpoly = base_ring.polynomial()(k.gen())
 
-        ch = base_ring.characteristic()
-        F = ch.factor()
-        assert(len(F)==1)
-
-        modbase = F[0][0]
-        cexponent = F[0][1]
-
         _ext_names = <char**>omAlloc0(sizeof(char*))
         _name = str_to_bytes(k._names[0])
         _ext_names[0] = omStrDup(_name)
-        _cfr = rDefault( modbase, 1, _ext_names )
+        _cfr = rDefault(<int>base_ring.characteristic(), 1, _ext_names)
 
         _cfr.qideal = idInit(1,1)
         rComplete(_cfr, 1)
@@ -422,60 +560,6 @@ cdef ring *singular_ring_new(base_ring, n, names, term_order) except NULL:
 
         _ring = rDefault (_cf, nvars, _names, nblcks, _order, _block0, _block1, _wvhdl)
 
-    elif isinstance(base_ring, sage.rings.abc.IntegerModRing):
-
-        ch = base_ring.characteristic()
-        if ch < 2:
-            raise NotImplementedError(f"polynomials over {base_ring} are not supported in Singular")
-
-        isprime = ch.is_prime()
-
-        if not isprime and ch.is_power_of(2):
-            exponent = ch.nbits() -1
-            cexponent = exponent
-
-            if exponent <= 30:
-                ringtype = n_Z2m
-            else:
-                ringtype = n_Znm
-
-            if ringtype == n_Znm:
-                F = ch.factor()
-
-                modbase = F[0][0]
-                cexponent = F[0][1]
-
-                _info.base = <__mpz_struct*>omAlloc(sizeof(__mpz_struct))
-                mpz_init_set_ui(_info.base, modbase)
-                _info.exp = cexponent
-                _cf = nInitChar(ringtype, <void *>&_info)
-            else:  # ringtype == n_Z2m
-                _cf = nInitChar(ringtype, <void *>cexponent)
-
-        elif not isprime and ch.is_prime_power() and ch < ZZ(2)**160:
-            F = ch.factor()
-            assert(len(F)==1)
-
-            modbase = F[0][0]
-            cexponent = F[0][1]
-
-            _info.base = <__mpz_struct*>omAlloc(sizeof(__mpz_struct))
-            mpz_init_set_ui(_info.base, modbase)
-            _info.exp = cexponent
-            _cf = nInitChar( n_Znm, <void *>&_info )
-
-        else:
-            try:
-                characteristic = ch
-            except OverflowError:
-                raise NotImplementedError("Characteristic %d too big." % ch)
-
-            _info.base = <__mpz_struct*>omAlloc(sizeof(__mpz_struct))
-            mpz_init_set_ui(_info.base, characteristic)
-            _info.exp = 1
-            _cf = nInitChar( n_Zn, <void *>&_info )
-        _ring = rDefault(_cf, nvars, _names, nblcks, _order, _block0, _block1, _wvhdl)
-
     else:
         raise NotImplementedError(f"polynomials over {base_ring} are not supported in Singular")
 

src/sage/libs/singular/singular.pxd

@@ -55,6 +55,13 @@ cdef number *sa2si_NF(object element, ring *_ring) noexcept
 # dispatches to all the above.
 cdef number *sa2si(Element elem, ring * _ring) noexcept
 
+# ==============
+# Error handling
+# ==============
+
+cdef int start_catch_error() except -1
+cdef object check_error()
+
 # ==============
 # Initialisation
 # ==============