diff --git a/mlir/include/mlir/Dialect/Polynomial/IR/Polynomial.h b/mlir/include/mlir/Dialect/Polynomial/IR/Polynomial.h
index 39b05b9d3ad14..3325a6fa3f9fc 100644
--- a/mlir/include/mlir/Dialect/Polynomial/IR/Polynomial.h
+++ b/mlir/include/mlir/Dialect/Polynomial/IR/Polynomial.h
@@ -51,9 +51,6 @@ class Monomial {
     return (exponent.ult(other.exponent));
   }
 
-  // Prints polynomial to 'os'.
-  void print(raw_ostream &os) const;
-
   friend ::llvm::hash_code hash_value(const Monomial &arg);
 
 public:
@@ -102,6 +99,8 @@ class Polynomial {
 
   unsigned getDegree() const;
 
+  ArrayRef<Monomial> getTerms() const { return terms; }
+
   friend ::llvm::hash_code hash_value(const Polynomial &arg);
 
 private:
diff --git a/mlir/include/mlir/Dialect/Polynomial/IR/Polynomial.td b/mlir/include/mlir/Dialect/Polynomial/IR/Polynomial.td
index 5d8da8399b01b..d3e3ac55677f8 100644
--- a/mlir/include/mlir/Dialect/Polynomial/IR/Polynomial.td
+++ b/mlir/include/mlir/Dialect/Polynomial/IR/Polynomial.td
@@ -35,18 +35,18 @@ def Polynomial_Dialect : Dialect {
 
     ```mlir
     // A constant polynomial in a ring with i32 coefficients and no polynomial modulus
-    #ring = #polynomial.ring<ctype=i32>
+    #ring = #polynomial.ring<coefficientType=i32>
     %a = polynomial.constant <1 + x**2 - 3x**3> : polynomial.polynomial<#ring>
 
     // A constant polynomial in a ring with i32 coefficients, modulo (x^1024 + 1)
     #modulus = #polynomial.polynomial<1 + x**1024>
-    #ring = #polynomial.ring<ctype=i32, ideal=#modulus>
+    #ring = #polynomial.ring<coefficientType=i32, polynomialModulus=#modulus>
     %a = polynomial.constant <1 + x**2 - 3x**3> : polynomial.polynomial<#ring>
 
     // A constant polynomial in a ring with i32 coefficients, with a polynomial
     // modulus of (x^1024 + 1) and a coefficient modulus of 17.
     #modulus = #polynomial.polynomial<1 + x**1024>
-    #ring = #polynomial.ring<ctype=i32, cmod=17, ideal=#modulus>
+    #ring = #polynomial.ring<coefficientType=i32, coefficientModulus=17, polynomialModulus=#modulus>
     %a = polynomial.constant <1 + x**2 - 3x**3> : polynomial.polynomial<#ring>
     ```
   }];
@@ -63,7 +63,21 @@ class Polynomial_Attr<string name, string attrMnemonic, list<Trait> traits = []>
 def Polynomial_PolynomialAttr : Polynomial_Attr<"Polynomial", "polynomial"> {
   let summary = "An attribute containing a single-variable polynomial.";
   let description = [{
-     #poly = #polynomial.poly<x**1024 + 1>
+    A polynomial attribute represents a single-variable polynomial, which
+    is used to define the modulus of a `RingAttr`, as well as to define constants
+    and perform constant folding for `polynomial` ops.
+
+    The polynomial must be expressed as a list of monomial terms, with addition
+    or subtraction between them. The choice of variable name is arbitrary, but
+    must be consistent across all the monomials used to define a single
+    attribute. The order of monomial terms is arbitrary, each monomial degree
+    must occur at most once.
+
+    Example:
+
+    ```mlir
+    #poly = #polynomial.polynomial<x**1024 + 1>
+    ```
   }];
   let parameters = (ins "Polynomial":$polynomial);
   let hasCustomAssemblyFormat = 1;
@@ -79,10 +93,10 @@ def Polynomial_RingAttr : Polynomial_Attr<"Ring", "ring"> {
     integral, whose coefficients are taken modulo some statically known modulus
     (`coefficientModulus`).
 
-    Additionally, a polynomial ring can specify an _ideal_, which converts
+    Additionally, a polynomial ring can specify a _polynomialModulus_, which converts
     polynomial arithmetic to the analogue of modular integer arithmetic, where
     each polynomial is represented as its remainder when dividing by the
-    modulus. For single-variable polynomials, an "ideal" is always specificed
+    modulus. For single-variable polynomials, an "polynomialModulus" is always specificed
     via a single polynomial, which we call `polynomialModulus`.
 
     An expressive example is polynomials with i32 coefficients, whose
@@ -122,32 +136,284 @@ class Polynomial_Type<string name, string typeMnemonic>
 
 def Polynomial_PolynomialType : Polynomial_Type<"Polynomial", "polynomial"> {
   let summary = "An element of a polynomial ring.";
-
   let description = [{
     A type for polynomials in a polynomial quotient ring.
   }];
-
   let parameters = (ins Polynomial_RingAttr:$ring);
   let assemblyFormat = "`<` $ring `>`";
 }
 
+def PolynomialLike: TypeOrContainer<Polynomial_PolynomialType, "polynomial-like">;
+
 class Polynomial_Op<string mnemonic, list<Trait> traits = []> :
-    Op<Polynomial_Dialect, mnemonic, traits # [Pure]>;
+    Op<Polynomial_Dialect, mnemonic, traits # [Pure]> {
+  let assemblyFormat = "operands attr-dict `:` functional-type(operands, results)";
+}
 
 class Polynomial_UnaryOp<string mnemonic, list<Trait> traits = []> :
     Polynomial_Op<mnemonic, traits # [SameOperandsAndResultType]> {
   let arguments = (ins Polynomial_PolynomialType:$operand);
   let results = (outs Polynomial_PolynomialType:$result);
-
-  let assemblyFormat = "$operand attr-dict `:` qualified(type($result))";
 }
 
 class Polynomial_BinaryOp<string mnemonic, list<Trait> traits = []> :
-    Polynomial_Op<mnemonic, traits # [SameOperandsAndResultType]> {
-  let arguments = (ins Polynomial_PolynomialType:$lhs, Polynomial_PolynomialType:$rhs);
-  let results = (outs Polynomial_PolynomialType:$result);
+    Polynomial_Op<mnemonic, !listconcat(traits, [Pure, SameOperandsAndResultType, ElementwiseMappable])> {
+  let arguments = (ins PolynomialLike:$lhs, PolynomialLike:$rhs);
+  let results = (outs PolynomialLike:$result);
+  let assemblyFormat = "operands attr-dict `:` type($result)";
+}
+
+def Polynomial_AddOp : Polynomial_BinaryOp<"add", [Commutative]> {
+  let summary = "Addition operation between polynomials.";
+  let description = [{
+    Performs polynomial addition on the operands. The operands may be single
+    polynomials or containers of identically-typed polynomials, i.e., polynomials
+    from the same underlying ring with the same coefficient types.
+
+    Addition is defined to occur in the ring defined by the ring attribute of
+    the two operands, meaning the addition is taken modulo the coefficientModulus
+    and the polynomialModulus of the ring.
+
+    Example:
+
+    ```mlir
+    // add two polynomials modulo x^1024 - 1
+    #poly = #polynomial.polynomial<x**1024 - 1>
+    #ring = #polynomial.ring<coefficientType=i32, coefficientModulus=65536, polynomialModulus=#poly>
+    %0 = polynomial.constant #polynomial.polynomial<1 + x**2> : !polynomial.polynomial<#ring>
+    %1 = polynomial.constant #polynomial.polynomial<x**5 - x + 1> : !polynomial.polynomial<#ring>
+    %2 = polynomial.add %0, %1 : !polynomial.polynomial<#ring>
+    ```
+  }];
+}
+
+def Polynomial_SubOp : Polynomial_BinaryOp<"sub"> {
+  let summary = "Subtraction operation between polynomials.";
+  let description = [{
+    Performs polynomial subtraction on the operands. The operands may be single
+    polynomials or containers of identically-typed polynomials, i.e., polynomials
+    from the same underlying ring with the same coefficient types.
+
+    Subtraction is defined to occur in the ring defined by the ring attribute of
+    the two operands, meaning the subtraction is taken modulo the coefficientModulus
+    and the polynomialModulus of the ring.
+
+    Example:
 
-  let assemblyFormat = "$lhs `,` $rhs attr-dict `:` qualified(type($result))";
+    ```mlir
+    // subtract two polynomials modulo x^1024 - 1
+    #poly = #polynomial.polynomial<x**1024 - 1>
+    #ring = #polynomial.ring<coefficientType=i32, coefficientModulus=65536, polynomialModulus=#poly>
+    %0 = polynomial.constant #polynomial.polynomial<1 + x**2> : !polynomial.polynomial<#ring>
+    %1 = polynomial.constant #polynomial.polynomial<x**5 - x + 1> : !polynomial.polynomial<#ring>
+    %2 = polynomial.sub %0, %1 : !polynomial.polynomial<#ring>
+    ```
+  }];
+}
+
+def Polynomial_MulOp : Polynomial_BinaryOp<"mul", [Commutative]> {
+  let summary = "Multiplication operation between polynomials.";
+  let description = [{
+    Performs polynomial multiplication on the operands. The operands may be single
+    polynomials or containers of identically-typed polynomials, i.e., polynomials
+    from the same underlying ring with the same coefficient types.
+
+    Multiplication is defined to occur in the ring defined by the ring attribute of
+    the two operands, meaning the multiplication is taken modulo the coefficientModulus
+    and the polynomialModulus of the ring.
+
+    Example:
+
+    ```mlir
+    // multiply two polynomials modulo x^1024 - 1
+    #poly = #polynomial.polynomial<x**1024 - 1>
+    #ring = #polynomial.ring<coefficientType=i32, coefficientModulus=65536, polynomialModulus=#poly>
+    %0 = polynomial.constant #polynomial.polynomial<1 + x**2> : !polynomial.polynomial<#ring>
+    %1 = polynomial.constant #polynomial.polynomial<x**5 - x + 1> : !polynomial.polynomial<#ring>
+    %2 = polynomial.mul %0, %1 : !polynomial.polynomial<#ring>
+    ```
+  }];
+}
+
+def Polynomial_MulScalarOp : Polynomial_Op<"mul_scalar", [
+      ElementwiseMappable, AllTypesMatch<["polynomial", "output"]>]> {
+  let summary = "Multiplication by a scalar of the field.";
+  let description = [{
+    Multiplies the polynomial operand's coefficients by a given scalar value.
+    The operation is defined to occur in the ring defined by the ring attribute
+    of the two operands, meaning the multiplication is taken modulo the
+    coefficientModulus of the ring.
+
+    The `scalar` input must have the same type as the polynomial ring's
+    coefficientType.
+
+    Example:
+
+    ```mlir
+    // multiply two polynomials modulo x^1024 - 1
+    #poly = #polynomial.polynomial<x**1024 - 1>
+    #ring = #polynomial.ring<coefficientType=i32, coefficientModulus=65536, polynomialModulus=#poly>
+    %0 = polynomial.constant #polynomial.polynomial<1 + x**2> : !polynomial.polynomial<#ring>
+    %1 = arith.constant 3 : i32
+    %2 = polynomial.mul_scalar %0, %1 : !polynomial.polynomial<#ring>, i32
+    ```
+  }];
+
+  let arguments = (ins
+    PolynomialLike:$polynomial,
+    AnyInteger:$scalar
+  );
+  let results = (outs
+    PolynomialLike:$output
+  );
+  let assemblyFormat = "operands attr-dict `:` type($polynomial) `,` type($scalar)";
+  let hasVerifier = 1;
+}
+
+def Polynomial_LeadingTermOp: Polynomial_Op<"leading_term"> {
+  let summary = "Compute the leading term of the polynomial.";
+  let description = [{
+    The degree of a polynomial is the largest $k$ for which the coefficient
+    `a_k` of `x^k` is nonzero. The leading term is the term `a_k * x^k`, which
+    this op represents as a pair of results. The first is the degree `k` as an
+    index, and the second is the coefficient, whose type matches the
+    coefficient type of the polynomial's ring attribute.
+
+    Example:
+
+    ```mlir
+    #poly = #polynomial.polynomial<x**1024 - 1>
+    #ring = #polynomial.ring<coefficientType=i32, coefficientModulus=65536, polynomialModulus=#poly>
+    %0 = polynomial.constant #polynomial.polynomial<1 + x**2> : !polynomial.polynomial<#ring>
+    %1, %2 = polynomial.leading_term %0 : !polynomial.polynomial<#ring> -> (index, i32)
+    ```
+  }];
+  let arguments = (ins Polynomial_PolynomialType:$input);
+  let results = (outs Index:$degree, AnyInteger:$coefficient);
+  let assemblyFormat = "operands attr-dict `:` type($input) `->` `(` type($degree) `,` type($coefficient) `)`";
+}
+
+def Polynomial_MonomialOp: Polynomial_Op<"monomial"> {
+  let summary = "Create a polynomial that consists of a single monomial.";
+  let description = [{
+    Construct a polynomial that consists of a single monomial term, from its
+    degree and coefficient as dynamic inputs.
+
+    The coefficient type of the output polynomial's ring attribute must match
+    the `coefficient` input type.
+
+    Example:
+
+    ```mlir
+    #poly = #polynomial.polynomial<x**1024 - 1>
+    #ring = #polynomial.ring<coefficientType=i32, coefficientModulus=65536, polynomialModulus=#poly>
+    %deg = arith.constant 1023 : index
+    %five = arith.constant 5 : i32
+    %0 = polynomial.monomial %five, %deg : (i32, index) -> !polynomial.polynomial<#ring>
+    ```
+  }];
+  let arguments = (ins AnyInteger:$coefficient, Index:$degree);
+  let results = (outs Polynomial_PolynomialType:$output);
+}
+
+def Polynomial_MonicMonomialMulOp: Polynomial_Op<"monic_monomial_mul", [AllTypesMatch<["input", "output"]>]> {
+  let summary = "Multiply a polynomial by a monic monomial.";
+  let description = [{
+    Multiply a polynomial by a monic monomial, meaning a polynomial of the form
+    `1 * x^k` for an index operand `k`.
+
+    In some special rings of polynomials, such as a ring of polynomials
+    modulo `x^n - 1`, `monomial_mul` can be interpreted as a cyclic shift of
+    the coefficients of the polynomial. For some rings, this results in
+    optimized lowerings that involve rotations and rescaling of the
+    coefficients of the input.
+  }];
+  let arguments = (ins PolynomialLike:$input, Index:$monomialDegree);
+  let results = (outs PolynomialLike:$output);
+}
+
+def Polynomial_FromTensorOp : Polynomial_Op<"from_tensor", [Pure]> {
+  let summary = "Creates a polynomial from integer coefficients stored in a tensor.";
+  let description = [{
+    `polynomial.from_tensor` creates a polynomial value from a tensor of coefficients.
+    The input tensor must list the coefficients in degree-increasing order.
+
+    The input one-dimensional tensor may have size at most the degree of the
+    ring's polynomialModulus generator polynomial, with smaller dimension implying that
+    all higher-degree terms have coefficient zero.
+
+    Example:
+
+    ```mlir
+    #poly = #polynomial.polynomial<x**1024 - 1>
+    #ring = #polynomial.ring<coefficientType=i32, coefficientModulus=65536, polynomialModulus=#poly>
+    %two = arith.constant 2 : i32
+    %five = arith.constant 5 : i32
+    %coeffs = tensor.from_elements %two, %two, %five : tensor<3xi32>
+    %poly = polynomial.from_tensor %coeffs : tensor<3xi32> -> !polynomial.polynomial<#ring>
+    ```
+  }];
+  let arguments = (ins RankedTensorOf<[AnyInteger]>:$input);
+  let results = (outs Polynomial_PolynomialType:$output);
+
+  let assemblyFormat = "$input attr-dict `:` type($input) `->` type($output)";
+
+  let builders = [
+    // Builder that infers coefficient modulus from tensor bit width,
+    // and uses whatever input ring is provided by the caller.
+    OpBuilder<(ins "::mlir::Value":$input, "::mlir::polynomial::RingAttr":$ring)>
+  ];
+  let hasVerifier = 1;
+}
+
+def Polynomial_ToTensorOp : Polynomial_Op<"to_tensor", [Pure]> {
+  let summary = "Creates a tensor containing the coefficients of a polynomial.";
+  let description = [{
+    `polynomial.to_tensor` creates a dense tensor value containing the
+    coefficients of the input polynomial. The output tensor contains the
+    coefficients in degree-increasing order.
+
+    Operations that act on the coefficients of a polynomial, such as extracting
+    a specific coefficient or extracting a range of coefficients, should be
+    implemented by composing `to_tensor` with the relevant `tensor` dialect
+    ops.
+
+    The output tensor has shape equal to the degree of the polynomial ring
+    attribute's polynomialModulus, including zeroes.
+
+    Example:
+
+    ```mlir
+    #poly = #polynomial.polynomial<x**1024 - 1>
+    #ring = #polynomial.ring<coefficientType=i32, coefficientModulus=65536, polynomialModulus=#poly>
+    %two = arith.constant 2 : i32
+    %five = arith.constant 5 : i32
+    %coeffs = tensor.from_elements %two, %two, %five : tensor<3xi32>
+    %poly = polynomial.from_tensor %coeffs : tensor<3xi32> -> !polynomial.polynomial<#ring>
+    %tensor = polynomial.to_tensor %poly : !polynomial.polynomial<#ring> -> tensor<1024xi32>
+    ```
+  }];
+  let arguments = (ins Polynomial_PolynomialType:$input);
+  let results = (outs RankedTensorOf<[AnyInteger]>:$output);
+  let assemblyFormat = "$input attr-dict `:` type($input) `->` type($output)";
+
+  let hasVerifier = 1;
+}
+
+def Polynomial_ConstantOp : Polynomial_Op<"constant", [Pure]> {
+  let summary = "Define a constant polynomial via an attribute.";
+  let description = [{
+    Example:
+
+    ```mlir
+    #poly = #polynomial.polynomial<x**1024 - 1>
+    #ring = #polynomial.ring<coefficientType=i32, coefficientModulus=65536, polynomialModulus=#poly>
+    %0 = polynomial.constant #polynomial.polynomial<1 + x**2> : !polynomial.polynomial<#ring>
+    ```
+  }];
+  let arguments = (ins Polynomial_PolynomialAttr:$input);
+  let results = (outs Polynomial_PolynomialType:$output);
+  let assemblyFormat = "$input attr-dict `:` type($output)";
 }
 
 #endif // POLYNOMIAL_OPS
diff --git a/mlir/lib/Dialect/Polynomial/IR/PolynomialDialect.cpp b/mlir/lib/Dialect/Polynomial/IR/PolynomialDialect.cpp
index a672a59b8a465..825b80d70f803 100644
--- a/mlir/lib/Dialect/Polynomial/IR/PolynomialDialect.cpp
+++ b/mlir/lib/Dialect/Polynomial/IR/PolynomialDialect.cpp
@@ -8,9 +8,18 @@
 
 #include "mlir/Dialect/Polynomial/IR/Polynomial.h"
 
+#include "mlir/Dialect/Arith/IR/Arith.h"
 #include "mlir/Dialect/Polynomial/IR/PolynomialAttributes.h"
 #include "mlir/Dialect/Polynomial/IR/PolynomialOps.h"
 #include "mlir/Dialect/Polynomial/IR/PolynomialTypes.h"
+#include "mlir/IR/Builders.h"
+#include "mlir/IR/BuiltinOps.h"
+#include "mlir/IR/BuiltinTypes.h"
+#include "mlir/IR/Dialect.h"
+#include "mlir/IR/PatternMatch.h"
+#include "mlir/Interfaces/InferTypeOpInterface.h"
+#include "mlir/Support/LogicalResult.h"
+#include "llvm/ADT/APInt.h"
 #include "llvm/ADT/TypeSwitch.h"
 
 using namespace mlir;
diff --git a/mlir/lib/Dialect/Polynomial/IR/PolynomialOps.cpp b/mlir/lib/Dialect/Polynomial/IR/PolynomialOps.cpp
index 96c59a28b8fdc..8e2bb5f27dc6c 100644
--- a/mlir/lib/Dialect/Polynomial/IR/PolynomialOps.cpp
+++ b/mlir/lib/Dialect/Polynomial/IR/PolynomialOps.cpp
@@ -6,10 +6,101 @@
 //
 //===----------------------------------------------------------------------===//
 
+#include "mlir/Dialect/Polynomial/IR/PolynomialOps.h"
 #include "mlir/Dialect/Polynomial/IR/Polynomial.h"
+#include "mlir/Dialect/Polynomial/IR/PolynomialAttributes.h"
+#include "mlir/Dialect/Polynomial/IR/PolynomialTypes.h"
+#include "mlir/IR/Builders.h"
+#include "mlir/IR/BuiltinTypes.h"
+#include "mlir/IR/Dialect.h"
+#include "mlir/Support/LogicalResult.h"
+#include "llvm/ADT/APInt.h"
 
 using namespace mlir;
 using namespace mlir::polynomial;
 
-#define GET_OP_CLASSES
-#include "mlir/Dialect/Polynomial/IR/Polynomial.cpp.inc"
+void FromTensorOp::build(OpBuilder &builder, OperationState &result,
+                         Value input, RingAttr ring) {
+  TensorType tensorType = dyn_cast<TensorType>(input.getType());
+  auto bitWidth = tensorType.getElementTypeBitWidth();
+  APInt cmod(1 + bitWidth, 1);
+  cmod = cmod << bitWidth;
+  Type resultType = PolynomialType::get(builder.getContext(), ring);
+  build(builder, result, resultType, input);
+}
+
+LogicalResult FromTensorOp::verify() {
+  ArrayRef<int64_t> tensorShape = getInput().getType().getShape();
+  RingAttr ring = getOutput().getType().getRing();
+  unsigned polyDegree = ring.getPolynomialModulus().getPolynomial().getDegree();
+  bool compatible = tensorShape.size() == 1 && tensorShape[0] <= polyDegree;
+  if (!compatible) {
+    InFlightDiagnostic diag = emitOpError()
+                              << "input type " << getInput().getType()
+                              << " does not match output type "
+                              << getOutput().getType();
+    diag.attachNote() << "the input type must be a tensor of shape [d] where d "
+                         "is at most the degree of the polynomialModulus of "
+                         "the output type's ring attribute";
+    return diag;
+  }
+
+  APInt coefficientModulus = ring.getCoefficientModulus().getValue();
+  unsigned cmodBitWidth = coefficientModulus.ceilLogBase2();
+  unsigned inputBitWidth = getInput().getType().getElementTypeBitWidth();
+
+  if (inputBitWidth > cmodBitWidth) {
+    InFlightDiagnostic diag = emitOpError()
+                              << "input tensor element type "
+                              << getInput().getType().getElementType()
+                              << " is too large to fit in the coefficients of "
+                              << getOutput().getType();
+    diag.attachNote() << "the input tensor's elements must be rescaled"
+                         " to fit before using from_tensor";
+    return diag;
+  }
+
+  return success();
+}
+
+LogicalResult ToTensorOp::verify() {
+  ArrayRef<int64_t> tensorShape = getOutput().getType().getShape();
+  unsigned polyDegree = getInput()
+                            .getType()
+                            .getRing()
+                            .getPolynomialModulus()
+                            .getPolynomial()
+                            .getDegree();
+  bool compatible = tensorShape.size() == 1 && tensorShape[0] == polyDegree;
+
+  if (compatible)
+    return success();
+
+  InFlightDiagnostic diag =
+      emitOpError() << "input type " << getInput().getType()
+                    << " does not match output type " << getOutput().getType();
+  diag.attachNote() << "the output type must be a tensor of shape [d] where d "
+                       "is at most the degree of the polynomialModulus of "
+                       "the input type's ring attribute";
+  return diag;
+}
+
+LogicalResult MulScalarOp::verify() {
+  Type argType = getPolynomial().getType();
+  PolynomialType polyType;
+
+  if (auto shapedPolyType = dyn_cast<ShapedType>(argType)) {
+    polyType = cast<PolynomialType>(shapedPolyType.getElementType());
+  } else {
+    polyType = cast<PolynomialType>(argType);
+  }
+
+  Type coefficientType = polyType.getRing().getCoefficientType();
+
+  if (coefficientType != getScalar().getType())
+    return emitOpError() << "polynomial coefficient type " << coefficientType
+                         << " does not match scalar type "
+                         << getScalar().getType();
+
+  return success();
+}
diff --git a/mlir/test/Dialect/Polynomial/ops.mlir b/mlir/test/Dialect/Polynomial/ops.mlir
new file mode 100644
index 0000000000000..ea1b279fa1ff9
--- /dev/null
+++ b/mlir/test/Dialect/Polynomial/ops.mlir
@@ -0,0 +1,82 @@
+// RUN: mlir-opt %s | FileCheck %s
+
+// This simply tests for syntax.
+
+#my_poly = #polynomial.polynomial<1 + x**1024>
+#my_poly_2 = #polynomial.polynomial<2>
+#my_poly_3 = #polynomial.polynomial<3x>
+#my_poly_4 = #polynomial.polynomial<t**3 + 4t + 2>
+#ring1 = #polynomial.ring<coefficientType=i32, coefficientModulus=2837465, polynomialModulus=#my_poly>
+#one_plus_x_squared = #polynomial.polynomial<1 + x**2>
+
+#ideal = #polynomial.polynomial<-1 + x**1024>
+#ring = #polynomial.ring<coefficientType=i32, coefficientModulus=18, polynomialModulus=#ideal>
+!poly_ty = !polynomial.polynomial<#ring>
+
+module {
+  func.func @test_multiply() -> !polynomial.polynomial<#ring1> {
+    %c0 = arith.constant 0 : index
+    %two = arith.constant 2 : i16
+    %five = arith.constant 5 : i16
+    %coeffs1 = tensor.from_elements %two, %two, %five : tensor<3xi16>
+    %coeffs2 = tensor.from_elements %five, %five, %two : tensor<3xi16>
+
+    %poly1 = polynomial.from_tensor %coeffs1 : tensor<3xi16> -> !polynomial.polynomial<#ring1>
+    %poly2 = polynomial.from_tensor %coeffs2 : tensor<3xi16> -> !polynomial.polynomial<#ring1>
+
+    %3 = polynomial.mul %poly1, %poly2 : !polynomial.polynomial<#ring1>
+
+    return %3 : !polynomial.polynomial<#ring1>
+  }
+
+  func.func @test_elementwise(%p0 : !polynomial.polynomial<#ring1>, %p1: !polynomial.polynomial<#ring1>) {
+    %tp0 = tensor.from_elements %p0, %p1 : tensor<2x!polynomial.polynomial<#ring1>>
+    %tp1 = tensor.from_elements %p1, %p0 : tensor<2x!polynomial.polynomial<#ring1>>
+
+    %c = arith.constant 2 : i32
+    %mul_const_sclr = polynomial.mul_scalar %tp0, %c : tensor<2x!polynomial.polynomial<#ring1>>, i32
+
+    %add = polynomial.add %tp0, %tp1 : tensor<2x!polynomial.polynomial<#ring1>>
+    %sub = polynomial.sub %tp0, %tp1 : tensor<2x!polynomial.polynomial<#ring1>>
+    %mul = polynomial.mul %tp0, %tp1 : tensor<2x!polynomial.polynomial<#ring1>>
+
+    return
+  }
+
+  func.func @test_to_from_tensor(%p0 : !polynomial.polynomial<#ring1>) {
+    %c0 = arith.constant 0 : index
+    %two = arith.constant 2 : i16
+    %coeffs1 = tensor.from_elements %two, %two : tensor<2xi16>
+    // CHECK: from_tensor
+    %poly = polynomial.from_tensor %coeffs1 : tensor<2xi16> -> !polynomial.polynomial<#ring1>
+    // CHECK: to_tensor
+    %tensor = polynomial.to_tensor %poly : !polynomial.polynomial<#ring1> -> tensor<1024xi16>
+
+    return
+  }
+
+  func.func @test_degree(%p0 : !polynomial.polynomial<#ring1>) {
+    %0, %1 = polynomial.leading_term %p0 : !polynomial.polynomial<#ring1> -> (index, i32)
+    return
+  }
+
+  func.func @test_monomial() {
+    %deg = arith.constant 1023 : index
+    %five = arith.constant 5 : i16
+    %0 = polynomial.monomial %five, %deg : (i16, index) -> !polynomial.polynomial<#ring1>
+    return
+  }
+
+  func.func @test_monic_monomial_mul() {
+    %five = arith.constant 5 : index
+    %0 = polynomial.constant #one_plus_x_squared : !polynomial.polynomial<#ring1>
+    %1 = polynomial.monic_monomial_mul %0, %five : (!polynomial.polynomial<#ring1>, index) -> !polynomial.polynomial<#ring1>
+    return
+  }
+
+  func.func @test_constant() {
+    %0 = polynomial.constant #one_plus_x_squared : !polynomial.polynomial<#ring1>
+    %1 = polynomial.constant <1 + x**2> : !polynomial.polynomial<#ring1>
+    return
+  }
+}
diff --git a/mlir/test/Dialect/Polynomial/ops_errors.mlir b/mlir/test/Dialect/Polynomial/ops_errors.mlir
new file mode 100644
index 0000000000000..c34a7de30e5fe
--- /dev/null
+++ b/mlir/test/Dialect/Polynomial/ops_errors.mlir
@@ -0,0 +1,53 @@
+// RUN: mlir-opt --split-input-file --verify-diagnostics %s
+
+#my_poly = #polynomial.polynomial<1 + x**1024>
+#ring = #polynomial.ring<coefficientType=i16, coefficientModulus=256, polynomialModulus=#my_poly>
+!ty = !polynomial.polynomial<#ring>
+
+func.func @test_from_tensor_too_large_coeffs() {
+  %two = arith.constant 2 : i32
+  %coeffs1 = tensor.from_elements %two, %two : tensor<2xi32>
+  // expected-error@below {{is too large to fit in the coefficients}}
+  // expected-note@below {{rescaled to fit}}
+  %poly = polynomial.from_tensor %coeffs1 : tensor<2xi32> -> !ty
+  return
+}
+
+// -----
+
+#my_poly = #polynomial.polynomial<1 + x**4>
+#ring = #polynomial.ring<coefficientType=i32, coefficientModulus=256, polynomialModulus=#my_poly>
+!ty = !polynomial.polynomial<#ring>
+func.func @test_from_tensor_wrong_tensor_type() {
+  %two = arith.constant 2 : i32
+  %coeffs1 = tensor.from_elements %two, %two, %two, %two, %two : tensor<5xi32>
+  // expected-error@below {{input type 'tensor<5xi32>' does not match output type '!polynomial.polynomial<#polynomial.ring<coefficientType=i32, coefficientModulus=256 : i32, polynomialModulus=#polynomial.polynomial<1 + x**4>>>'}}
+  // expected-note@below {{at most the degree of the polynomialModulus of the output type's ring attribute}}
+  %poly = polynomial.from_tensor %coeffs1 : tensor<5xi32> -> !ty
+  return
+}
+
+// -----
+
+#my_poly = #polynomial.polynomial<1 + x**4>
+#ring = #polynomial.ring<coefficientType=i32, coefficientModulus=256, polynomialModulus=#my_poly>
+!ty = !polynomial.polynomial<#ring>
+func.func @test_to_tensor_wrong_output_tensor_type(%arg0 : !ty) {
+  // expected-error@below {{input type '!polynomial.polynomial<#polynomial.ring<coefficientType=i32, coefficientModulus=256 : i32, polynomialModulus=#polynomial.polynomial<1 + x**4>>>' does not match output type 'tensor<5xi32>'}}
+  // expected-note@below {{at most the degree of the polynomialModulus of the input type's ring attribute}}
+  %tensor = polynomial.to_tensor %arg0 : !ty -> tensor<5xi32>
+  return
+}
+
+// -----
+
+#my_poly = #polynomial.polynomial<1 + x**1024>
+#ring = #polynomial.ring<coefficientType=i16, coefficientModulus=256, polynomialModulus=#my_poly>
+!ty = !polynomial.polynomial<#ring>
+
+func.func @test_mul_scalar_wrong_type(%arg0: !ty) -> !ty {
+  %scalar = arith.constant 2 : i32  // should be i16
+  // expected-error@below {{polynomial coefficient type 'i16' does not match scalar type 'i32'}}
+  %poly = polynomial.mul_scalar %arg0, %scalar : !ty, i32
+  return %poly : !ty
+}
diff --git a/mlir/test/Dialect/Polynomial/types.mlir b/mlir/test/Dialect/Polynomial/types.mlir
index 64b74d9d36bb1..00296a36e890f 100644
--- a/mlir/test/Dialect/Polynomial/types.mlir
+++ b/mlir/test/Dialect/Polynomial/types.mlir
@@ -40,3 +40,17 @@ func.func @test_non_x_variable_64_bit(%0: !ty2) -> !ty2 {
 func.func @test_linear_poly(%0: !ty3) -> !ty3 {
   return %0 : !ty3
 }
+
+// CHECK-LABEL: func @test_negative_leading_1
+// CHECK-SAME:  !polynomial.polynomial<
+// CHECK-SAME:    #polynomial.ring<
+// CHECK-SAME:       coefficientType=i32,
+// CHECK-SAME:       coefficientModulus=2837465 : i32,
+// CHECK-SAME:       polynomialModulus=#polynomial.polynomial<-1 + x**1024>>>
+#my_poly_4 = #polynomial.polynomial<-1 + x**1024>
+#ring4 = #polynomial.ring<coefficientType=i32, coefficientModulus=2837465, polynomialModulus=#my_poly_4>
+!ty4 = !polynomial.polynomial<#ring4>
+func.func @test_negative_leading_1(%0: !ty4) -> !ty4 {
+  return %0 : !ty4
+}
+