diff --git a/crates/prover/src/constraint_framework/mod.rs b/crates/prover/src/constraint_framework/mod.rs
index 377dcdefb..cbc7eb836 100644
--- a/crates/prover/src/constraint_framework/mod.rs
+++ b/crates/prover/src/constraint_framework/mod.rs
@@ -6,6 +6,7 @@ mod cpu_domain;
 mod info;
 pub mod logup;
 mod point;
+pub mod poly;
 mod simd_domain;
 
 use std::array;
diff --git a/crates/prover/src/constraint_framework/poly.rs b/crates/prover/src/constraint_framework/poly.rs
new file mode 100644
index 000000000..5f8dcfa61
--- /dev/null
+++ b/crates/prover/src/constraint_framework/poly.rs
@@ -0,0 +1,366 @@
+use std::collections::BTreeMap;
+use std::fmt::{Display, Formatter};
+use std::ops::{Add, AddAssign, Mul, Neg};
+
+use itertools::Itertools;
+use num_traits::{One, Zero};
+
+use crate::core::fields::m31::BaseField;
+use crate::core::fields::qm31::SecureField;
+
+/// A monic monomial consists of a list of variables and their exponents.
+#[derive(Debug, Hash, PartialEq, PartialOrd, Eq, Ord, Clone)]
+pub struct Monomial {
+    /// The variables of the monomial and their exponents.
+    pub vars: BTreeMap<usize, usize>,
+}
+
+impl Monomial {
+    pub fn degree(&self) -> usize {
+        self.vars.values().sum()
+    }
+
+    fn default() -> Monomial {
+        Monomial {
+            vars: [(0, 0)].into(),
+        }
+    }
+}
+
+/// A polynomial consists of a list of monomials with coefficients.
+#[derive(Debug, Hash, PartialEq, PartialOrd, Eq, Ord, Clone)]
+struct Polynomial<F: From<BaseField>> {
+    monomials: BTreeMap<Monomial, F>,
+}
+
+impl<F: One + From<BaseField>> From<Monomial> for Polynomial<F> {
+    fn from(monomial: Monomial) -> Self {
+        Self {
+            monomials: [(monomial, F::one())].into(),
+        }
+    }
+}
+
+impl<F: From<BaseField>> From<F> for Polynomial<F>
+where
+    F: One + Clone,
+{
+    fn from(scalar: F) -> Self {
+        Self {
+            monomials: [(Monomial::default(), scalar)].into(),
+        }
+    }
+}
+
+impl<F: Zero + Add + Clone + From<BaseField>> Add for Polynomial<F> {
+    type Output = Self;
+    fn add(self, rhs: Self) -> Self {
+        let mut monomials = self.monomials;
+        for (monomial, coef) in rhs.monomials {
+            if let Some(existing) = monomials.get_mut(&monomial) {
+                let res = existing.clone() + coef;
+                if res.is_zero() {
+                    monomials.remove(&monomial);
+                } else {
+                    *existing = res;
+                }
+            } else {
+                monomials.insert(monomial, coef);
+            }
+        }
+        Self { monomials }
+    }
+}
+
+#[allow(clippy::suspicious_arithmetic_impl)]
+impl Mul for Monomial {
+    type Output = Self;
+    fn mul(self, rhs: Self) -> Self {
+        let mut vars = self.vars;
+        for (var, exp) in rhs.vars {
+            if let Some(existing) = vars.get_mut(&var) {
+                *existing += exp;
+            } else {
+                vars.insert(var, exp);
+            }
+        }
+        Monomial { vars }
+    }
+}
+
+impl<F> Mul for Polynomial<F>
+where
+    F: Clone + Mul<Output = F> + Add<Output = F> + AddAssign + From<BaseField>,
+{
+    type Output = Self;
+    fn mul(self, rhs: Self) -> Self {
+        let mut monomials = BTreeMap::new();
+        for (monomial1, coef1) in self.monomials {
+            for (monomial2, coef2) in rhs.monomials.clone() {
+                let monomial = monomial1.clone() * monomial2.clone();
+                let coef = coef1.clone() * coef2.clone();
+                if let Some(existing) = monomials.get_mut(&monomial) {
+                    *existing += coef;
+                } else {
+                    monomials.insert(monomial, coef);
+                }
+            }
+        }
+        Self { monomials }
+    }
+}
+
+impl<F> Mul<BaseField> for Polynomial<F>
+where
+    F: Clone + Mul<Output = F> + Add<Output = F> + AddAssign + From<BaseField>,
+{
+    type Output = Self;
+    fn mul(self, rhs: BaseField) -> Self {
+        Self {
+            monomials: self
+                .monomials
+                .into_iter()
+                .map(|(m, c)| (m, c * rhs.into()))
+                .collect(),
+        }
+    }
+}
+
+impl<F> Mul<SecureField> for Polynomial<F>
+where
+    F: Clone
+        + Mul<SecureField, Output = SecureField>
+        + Add<Output = F>
+        + AddAssign
+        + From<BaseField>,
+{
+    type Output = Polynomial<SecureField>;
+    fn mul(self, rhs: SecureField) -> Self::Output {
+        Self::Output {
+            monomials: self
+                .monomials
+                .into_iter()
+                .map(|(m, c)| (m, c * rhs))
+                .collect(),
+        }
+    }
+}
+
+impl<F: Zero + Clone + From<BaseField>> Zero for Polynomial<F> {
+    fn is_zero(&self) -> bool {
+        self.monomials.is_empty()
+    }
+    fn zero() -> Self {
+        Self {
+            monomials: BTreeMap::new(),
+        }
+    }
+}
+
+impl<F: Neg<Output = F> + From<BaseField>> Neg for Polynomial<F> {
+    type Output = Self;
+    fn neg(self) -> Self {
+        Self {
+            monomials: self
+                .monomials
+                .into_iter()
+                .map(|(m, c)| (m, c.neg()))
+                .collect(),
+        }
+    }
+}
+
+impl<F, G> AddAssign<G> for Polynomial<F>
+where
+    F: Clone + Add<Output = F> + Zero + From<BaseField>,
+    G: Into<Polynomial<F>>,
+{
+    fn add_assign(&mut self, rhs: G) {
+        self.monomials = (self.clone() + rhs.into()).monomials;
+    }
+}
+
+fn to_superscript(n: usize) -> String {
+    if n == 0 {
+        return std::char::from_u32(0x2070_u32).unwrap().to_string();
+    }
+    let mut n = n;
+    let mut res = String::new();
+    while n > 0 {
+        let rem = n % 10;
+        res.push(match rem {
+            1 => std::char::from_u32(0x00b9).unwrap(),
+            2 => std::char::from_u32(0x00b2).unwrap(),
+            3 => std::char::from_u32(0x00b3).unwrap(),
+            _ => std::char::from_u32(0x2070 + rem as u32).unwrap(),
+        });
+        n /= 10;
+    }
+    res.chars().rev().collect()
+}
+
+fn to_subscript(n: usize) -> String {
+    if n == 0 {
+        return std::char::from_u32(0x2080_u32).unwrap().to_string();
+    }
+    let mut n = n;
+    let mut res = String::new();
+    while n > 0 {
+        let rem = n % 10;
+        res.push(std::char::from_u32(0x2080 + rem as u32).unwrap());
+        n /= 10;
+    }
+    res.chars().rev().collect()
+}
+
+impl Display for Monomial {
+    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
+        for (var, exp) in self.vars.iter().sorted() {
+            write!(
+                f,
+                "x{}{}",
+                to_subscript(*var),
+                if *exp == 1 {
+                    "".to_string()
+                } else {
+                    to_superscript(*exp)
+                }
+            )?;
+        }
+        write!(f, "")
+    }
+}
+
+impl<F> Display for Polynomial<F>
+where
+    F: Display + Zero + Add + Clone + From<BaseField>,
+{
+    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
+        let mut is_first = true;
+        for (monomial, coef) in self.monomials.iter() {
+            if !coef.is_zero() {
+                if !is_first {
+                    write!(f, " + ")?;
+                } else {
+                    is_first = false;
+                }
+                write!(f, "{}", coef)?;
+                if !monomial.vars.is_empty() {
+                    write!(f, "{}", monomial)?;
+                }
+            }
+        }
+        write!(f, "")
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::{Monomial, Polynomial};
+    use crate::core::fields::m31::M31;
+
+    #[test]
+    fn test_polynomial_display() {
+        let monomial1 = Monomial {
+            vars: (0..10).map(|i| (i, i + 1)).collect(),
+        };
+        let monomial2 = Monomial {
+            vars: (0..10).map(|i| (i + 1, i + 1)).collect(),
+        };
+        let polynomial = Polynomial {
+            monomials: [(monomial1.clone(), M31(18)), (monomial2.clone(), M31(3))].into(),
+        };
+        assert_eq!(
+            polynomial.to_string(),
+            "18x₀x₁²x₂³x₃⁴x₄⁵x₅⁶x₆⁷x₇⁸x₈⁹x₉¹⁰ + 3x₁x₂²x₃³x₄⁴x₅⁵x₆⁶x₇⁷x₈⁸x₉⁹x₁₀¹⁰"
+        )
+    }
+
+    #[test]
+    fn test_add_polynomials() {
+        let monomial1 = Monomial {
+            vars: [(0, 1), (1, 2)].into(),
+        };
+        let monomial2 = Monomial {
+            vars: [(2, 1), (3, 2)].into(),
+        };
+        let monomial3 = Monomial {
+            vars: [(4, 1), (5, 2)].into(),
+        };
+        let monomial4 = Monomial {
+            vars: [(6, 1), (7, 2)].into(),
+        };
+        let poly1 = Polynomial::<M31> {
+            monomials: [
+                (monomial1.clone(), M31(1)),
+                (monomial2.clone(), M31(2)),
+                (monomial3.clone(), M31(8)),
+            ]
+            .into(),
+        };
+        let poly2 = Polynomial::<M31> {
+            monomials: [
+                (monomial2.clone(), M31(5)),
+                (monomial3.clone(), -M31(8)),
+                (monomial4.clone(), M31(6)),
+            ]
+            .into(),
+        };
+
+        assert_eq!(
+            (poly1.clone() + poly2.clone()).monomials,
+            [
+                (monomial1.clone(), M31(1)),
+                (monomial2.clone(), M31(7)),
+                (monomial4.clone(), M31(6))
+            ]
+            .into()
+        );
+    }
+
+    #[test]
+    fn test_mul_monomials() {
+        let monomial1 = Monomial {
+            vars: [(0, 1), (1, 2)].into(),
+        };
+        let monomial2 = Monomial {
+            vars: [(1, 1), (2, 2)].into(),
+        };
+        assert_eq!(
+            monomial1.clone() * monomial2.clone(),
+            Monomial {
+                vars: [(0, 1), (1, 3), (2, 2)].into()
+            }
+        );
+    }
+
+    #[test]
+    fn test_mul_polynomials() {
+        let monomial1 = Monomial {
+            vars: [(0, 1), (1, 2)].into(),
+        };
+        let monomial2 = Monomial {
+            vars: [(2, 1), (3, 2)].into(),
+        };
+        let monomial3 = Monomial {
+            vars: [(4, 1), (5, 2)].into(),
+        };
+        let poly1 = Polynomial::<M31> {
+            monomials: [(monomial1.clone(), M31(1)), (monomial2.clone(), M31(2))].into(),
+        };
+        let poly2 = Polynomial::<M31> {
+            monomials: [(monomial2.clone(), M31(5)), (monomial3.clone(), -M31(8))].into(),
+        };
+
+        assert_eq!(
+            (poly1.clone() * poly2.clone()).monomials,
+            [
+                (monomial1.clone() * monomial2.clone(), M31(5)),
+                (monomial1.clone() * monomial3.clone(), -M31(8)),
+                (monomial2.clone() * monomial2.clone(), M31(10)),
+                (monomial2.clone() * monomial3.clone(), -M31(16))
+            ]
+            .into()
+        );
+    }
+}