A polynomial is not answered with no roots

Sources/.editorconfig

@@ -57,6 +57,9 @@ file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed
 [Tests/UnitTests/Convenience/SpellingsAtRiskTest.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 
+[Tests/UnitTests/Algebra/SolveTest/APolynomialHasARootTest.cs]
+file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
+
 [Tests/UnitTests/Convenience/RoundIsExactTest.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 

Sources/AngouriMath/Functions/Continuous/Solvers/EquationSolver/SolveStatement.cs

@@ -42,9 +42,32 @@ private static Entity Minus(Entity left, Entity right)
         /// equation as the caller wrote it.
         /// </remarks>
         internal static Set WithoutSpuriousRoots(Set roots, Entity equation, Variable x)
-            => roots is FiniteSet finite && finite.Any(root => IsSpurious(equation, x, root))
-                ? finite.Where(root => !IsSpurious(equation, x, root)).ToSet()
-                : roots;
+        {
+            if (roots is not FiniteSet finite || !finite.Any(root => IsSpurious(equation, x, root)))
+                return roots;
+            var kept = finite.Where(root => !IsSpurious(equation, x, root)).ToSet();
+            // A polynomial of degree one or more has a root over the complex numbers, so where every
+            // candidate for one fails, what failed is the solver, and the empty set left would claim
+            // there is no root: (x + 1)^3 = 2 was { } while Cardano's C was 0 for a shifted cube, and
+            // 31x^4 - 62x^3 + 33x^2 - 2x - 1 = 0 while a biquadratic quartic was shifted by -b a / 4.
+            // Left unsolved instead, which says only that the roots were not found.
+            if (kept.Count == 0 && IsAPolynomialWithARoot(equation, x))
+                return new ConditionalSet(x, equation is Minusf(var left, var right) ? left.Equalizes(right) : equation.Equalizes(0));
+            return kept;
+        }
+
+        /// <summary>
+        /// Whether <paramref name="equation"/> is a polynomial in <paramref name="x"/> with numbers
+        /// for coefficients and a degree of one or more, read over the complex numbers, where such a
+        /// polynomial has a root.
+        /// </summary>
+        private static bool IsAPolynomialWithARoot(Entity equation, Variable x)
+            => MathS.Settings.Codomain.Value != Core.Domain.Real
+                && TreeAnalyzer.TryGetPolynomial(equation, x, out var monomials)
+                && monomials.Keys.All(power => power.Sign >= 0)
+                && monomials.Where(monomial => monomial.Key.Sign > 0)
+                    .Any(monomial => monomial.Value.Evaled is Number.Complex { IsZero: false })
+                && monomials.Values.All(coefficient => coefficient.Evaled is Number.Complex { IsFinite: true });
 
         /// <summary>
         /// Answers the condition itself where a root denies the independence the calculus

Sources/Tests/UnitTests/Algebra/SolveTest/APolynomialHasARootTest.cs

@@ -0,0 +1,53 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using AngouriMath.Core;
+using AngouriMath.Extensions;
+using AngouriMath.Functions.Algebra.AnalyticalSolving;
+using Xunit;
+using static AngouriMath.Entity;
+using static AngouriMath.Entity.Set;
+
+namespace AngouriMath.Tests.Algebra
+{
+    /// <summary>
+    /// A polynomial of degree one or more has a root over the complex numbers, so where every
+    /// candidate a solver found for one fails its check, the solver failed, and the empty set
+    /// would claim there is no root. <c>(x + 1)^3 = 2</c> and
+    /// <c>31x^4 - 62x^3 + 33x^2 - 2x - 1 = 0</c> were answered <c>{ }</c> that way. Asked here of
+    /// the check itself, with candidates that fail, since no polynomial reaches it so any more.
+    /// </summary>
+    [Trait("Area", "Algebra")]
+    public sealed class APolynomialHasARootTest
+    {
+        [Theory]
+        [InlineData("x^2 - 2")]
+        [InlineData("(x + 1)^3 - 2")]
+        [InlineData("31x^4 - 62x^3 + 33x^2 - 2x - 1")]
+        public void APolynomialWhoseCandidatesAllFailIsLeftUnsolved(string polynomial)
+            => Assert.IsType<ConditionalSet>(
+                StatementSolver.WithoutSpuriousRoots(new FiniteSet((Entity)5, (Entity)7), polynomial.ToEntity(), "x"));
+
+        /// <summary>The roots that hold are kept, as before.</summary>
+        [Fact]
+        public void TheRootsThatHoldAreKept()
+            => Assert.Equal("{ 2 }".ToEntity(),
+                StatementSolver.WithoutSpuriousRoots(new FiniteSet((Entity)2, (Entity)5), "x^2 - 4".ToEntity(), "x").InnerSimplified);
+
+        /// <summary>
+        /// An equation that is not a polynomial can have no root, and its empty set is an answer:
+        /// <c>sqrt(x) = -1</c>. So can a polynomial over the reals: <c>x^2 + 1 = 0</c>.
+        /// </summary>
+        [Fact]
+        public void TheOthersCanHaveNoRoot()
+        {
+            Assert.Equal(Empty, StatementSolver.WithoutSpuriousRoots(new FiniteSet((Entity)1), "sqrt(x) + 1".ToEntity(), "x"));
+            using var _ = MathS.Settings.Codomain.Set(Domain.Real);
+            Assert.Equal(Empty, StatementSolver.WithoutSpuriousRoots(new FiniteSet((Entity)5), "x^2 + 1".ToEntity(), "x"));
+        }
+    }
+}