diff --git a/Cargo.lock b/Cargo.lock
index b4c4e4745e..8a7d8c1cd9 100644
--- a/Cargo.lock
+++ b/Cargo.lock
@@ -1839,6 +1839,7 @@ name = "mvpoly"
 version = "0.1.0"
 dependencies = [
  "ark-ff",
+ "kimchi",
  "log",
  "mina-curves",
  "num-integer",
diff --git a/mvpoly/Cargo.toml b/mvpoly/Cargo.toml
index 197d70a76a..3fc904c90e 100644
--- a/mvpoly/Cargo.toml
+++ b/mvpoly/Cargo.toml
@@ -14,6 +14,7 @@ path = "src/lib.rs"
 
 [dependencies]
 ark-ff.workspace = true
+kimchi.workspace = true
 log.workspace = true
 num-integer.workspace = true
 o1-utils.workspace = true
diff --git a/mvpoly/src/prime.rs b/mvpoly/src/prime.rs
index a382c5a86b..35cecda903 100644
--- a/mvpoly/src/prime.rs
+++ b/mvpoly/src/prime.rs
@@ -148,6 +148,13 @@ use std::{
 };
 
 use ark_ff::{One, PrimeField, Zero};
+use kimchi::circuits::{
+    expr::{
+        ChallengeTerm, ConstantExpr, ConstantExprInner, ConstantTerm, Expr, ExprInner, Operations,
+        Variable,
+    },
+    gate::CurrOrNext,
+};
 use num_integer::binomial;
 use o1_utils::FieldHelpers;
 use rand::RngCore;
@@ -331,6 +338,37 @@ impl<F: PrimeField, const N: usize, const D: usize> Dense<F, N, D> {
         let coeffs = self.coeff.iter().map(|coef| *coef * c).collect();
         Self::from_coeffs(coeffs)
     }
+
+    /// Evaluate the polynomial at the vector point `x`.
+    ///
+    /// This is a dummy implementation. A cache can be used for the monomials to
+    /// speed up the computation.
+    pub fn eval(&self, x: &[F; N]) -> F {
+        let mut prime_gen = PrimeNumberGenerator::new();
+        let primes = prime_gen.get_first_nth_primes(N);
+        self.coeff
+            .iter()
+            .enumerate()
+            .fold(F::zero(), |acc, (i, c)| {
+                if i == 0 {
+                    acc + c
+                } else {
+                    let normalized_index = self.normalized_indices[i];
+                    // IMPROVEME: we should keep the prime decomposition somewhere.
+                    // It can be precomputed for a few multi-variate polynomials
+                    // vector space
+                    let prime_decomposition = naive_prime_factors(normalized_index, &mut prime_gen);
+                    let mut monomial = F::one();
+                    prime_decomposition.iter().for_each(|(p, d)| {
+                        // IMPROVEME: we should keep the inverse indices
+                        let inv_p = primes.iter().position(|&x| x == *p).unwrap();
+                        let x_p = x[inv_p].pow([*d as u64]);
+                        monomial *= x_p;
+                    });
+                    acc + *c * monomial
+                }
+            })
+    }
 }
 
 impl<F: PrimeField, const N: usize, const D: usize> Default for Dense<F, N, D> {
@@ -554,4 +592,138 @@ impl<F: PrimeField, const N: usize, const D: usize> From<F> for Dense<F, N, D> {
     }
 }
 
+impl<F: PrimeField, const N: usize, const D: usize> From<ConstantExprInner<F>> for Dense<F, N, D> {
+    fn from(expr: ConstantExprInner<F>) -> Self {
+        match expr {
+            // The unimplemented methods might be implemented in the future if
+            // we move to the challenge into the type of the constant
+            // terms/expressions
+            // Unrolling for visibility
+            ConstantExprInner::Challenge(ChallengeTerm::Alpha) => {
+                unimplemented!("The challenge alpha is not supposed to be used in this context")
+            }
+            ConstantExprInner::Challenge(ChallengeTerm::Beta) => {
+                unimplemented!("The challenge beta is not supposed to be used in this context")
+            }
+            ConstantExprInner::Challenge(ChallengeTerm::Gamma) => {
+                unimplemented!("The challenge gamma is not supposed to be used in this context")
+            }
+            ConstantExprInner::Challenge(ChallengeTerm::JointCombiner) => {
+                unimplemented!(
+                    "The challenge joint combiner is not supposed to be used in this context"
+                )
+            }
+            ConstantExprInner::Constant(ConstantTerm::EndoCoefficient) => {
+                unimplemented!(
+                    "The constant EndoCoefficient is not supposed to be used in this context"
+                )
+            }
+            ConstantExprInner::Constant(ConstantTerm::Mds {
+                row: _row,
+                col: _col,
+            }) => {
+                unimplemented!("The constant Mds is not supposed to be used in this context")
+            }
+            ConstantExprInner::Constant(ConstantTerm::Literal(c)) => Dense::from(c),
+        }
+    }
+}
+
+impl<F: PrimeField, const N: usize, const D: usize> From<Operations<ConstantExprInner<F>>>
+    for Dense<F, N, D>
+{
+    fn from(op: Operations<ConstantExprInner<F>>) -> Self {
+        use kimchi::circuits::expr::Operations::*;
+        match op {
+            Atom(op_const) => Self::from(op_const),
+            Add(c1, c2) => Self::from(*c1) + Self::from(*c2),
+            Sub(c1, c2) => Self::from(*c1) - Self::from(*c2),
+            Mul(c1, c2) => Self::from(*c1) * Self::from(*c2),
+            Square(c) => Self::from(*c.clone()) * Self::from(*c),
+            Double(c1) => Self::from(*c1).double(),
+            Pow(c, e) => {
+                // FIXME: dummy implementation
+                let p = Dense::from(*c);
+                let mut result = p.clone();
+                for _ in 0..e {
+                    result = result.clone() * p.clone();
+                }
+                result
+            }
+            Cache(_c, _) => {
+                unimplemented!("The module prime is supposed to be used for generic multivariate expressions, not tied to a specific use case like Kimchi with this constructor")
+            }
+            IfFeature(_c, _t, _f) => {
+                unimplemented!("The module prime is supposed to be used for generic multivariate expressions, not tied to a specific use case like Kimchi with this constructor")
+            }
+        }
+    }
+}
+
+impl<Column: Into<usize>, F: PrimeField, const N: usize, const D: usize>
+    From<Expr<ConstantExpr<F>, Column>> for Dense<F, N, D>
+{
+    fn from(expr: Expr<ConstantExpr<F>, Column>) -> Self {
+        // This is a dummy implementation
+        // TODO: Implement the actual conversion logic
+        use kimchi::circuits::expr::Operations::*;
+
+        match expr {
+            Atom(op_const) => {
+                match op_const {
+                    ExprInner::UnnormalizedLagrangeBasis(_) => {
+                        unimplemented!("Not used in this context")
+                    }
+                    ExprInner::VanishesOnZeroKnowledgeAndPreviousRows => {
+                        unimplemented!("Not used in this context")
+                    }
+                    ExprInner::Constant(c) => Self::from(c),
+                    ExprInner::Cell(Variable { col, row }) => {
+                        assert_eq!(row, CurrOrNext::Curr, "Only current row is supported for now. You cannot reference the next row");
+                        Self::from_variable(col)
+                    }
+                }
+            }
+            Add(e1, e2) => {
+                let p1 = Dense::from(*e1);
+                let p2 = Dense::from(*e2);
+                p1 + p2
+            }
+            Sub(e1, e2) => {
+                let p1 = Dense::from(*e1);
+                let p2 = Dense::from(*e2);
+                p1 - p2
+            }
+            Mul(e1, e2) => {
+                let p1 = Dense::from(*e1);
+                let p2 = Dense::from(*e2);
+                p1 * p2
+            }
+            Double(p) => {
+                let p = Dense::from(*p);
+                p.double()
+            }
+            Square(p) => {
+                let p = Dense::from(*p);
+                p.clone() * p.clone()
+            }
+            Pow(c, e) => {
+                // FIXME: dummy implementation
+                let p = Dense::from(*c);
+                let mut result = p.clone();
+                for _ in 0..e {
+                    result = result.clone() * p.clone();
+                }
+                result
+            }
+            Cache(_c, _) => {
+                unimplemented!("The module prime is supposed to be used for generic multivariate expressions, not tied to a specific use case like Kimchi with this constructor")
+            }
+            IfFeature(_c, _t, _f) => {
+                unimplemented!("The module prime is supposed to be used for generic multivariate expressions, not tied to a specific use case like Kimchi with this constructor")
+            }
+        }
+    }
+}
+
 // TODO: implement From/To Expr<F, Column>
diff --git a/mvpoly/tests/prime.rs b/mvpoly/tests/prime.rs
index a9211b3fff..fce3a84bfd 100644
--- a/mvpoly/tests/prime.rs
+++ b/mvpoly/tests/prime.rs
@@ -354,3 +354,102 @@ fn test_from_variable_column() {
     assert_eq!(p[3], Fp::zero());
     assert_eq!(p[4], Fp::one());
 }
+
+#[test]
+fn test_evaluation_zero_polynomial() {
+    let mut rng = o1_utils::tests::make_test_rng(None);
+
+    let random_evaluation: [Fp; 4] = std::array::from_fn(|_| Fp::rand(&mut rng));
+    let zero = Dense::<Fp, 4, 5>::zero();
+    let evaluation = zero.eval(&random_evaluation);
+    assert_eq!(evaluation, Fp::zero());
+}
+
+#[test]
+fn test_evaluation_constant_polynomial() {
+    let mut rng = o1_utils::tests::make_test_rng(None);
+
+    let random_evaluation: [Fp; 4] = std::array::from_fn(|_| Fp::rand(&mut rng));
+    let cst = Fp::rand(&mut rng);
+    let zero = Dense::<Fp, 4, 5>::from(cst);
+    let evaluation = zero.eval(&random_evaluation);
+    assert_eq!(evaluation, cst);
+}
+
+#[test]
+fn test_evaluation_predefined_polynomial() {
+    // Evaluating at random points
+    let mut rng = o1_utils::tests::make_test_rng(None);
+
+    let random_evaluation: [Fp; 2] = std::array::from_fn(|_| Fp::rand(&mut rng));
+    // P(X1, X2) = 2 + 3X1 + 4X2 + 5X1^2 + 6X1 X2 + 7 X2^2
+    let p = Dense::<Fp, 2, 2>::from_coeffs(vec![
+        Fp::from(2_u32),
+        Fp::from(3_u32),
+        Fp::from(4_u32),
+        Fp::from(5_u32),
+        Fp::from(6_u32),
+        Fp::from(7_u32),
+    ]);
+    let exp_eval = Fp::from(2_u32)
+        + Fp::from(3_u32) * random_evaluation[0]
+        + Fp::from(4_u32) * random_evaluation[1]
+        + Fp::from(5_u32) * random_evaluation[0] * random_evaluation[0]
+        + Fp::from(6_u32) * random_evaluation[0] * random_evaluation[1]
+        + Fp::from(7_u32) * random_evaluation[1] * random_evaluation[1];
+    let evaluation = p.eval(&random_evaluation);
+    assert_eq!(evaluation, exp_eval);
+}
+
+#[test]
+fn test_eval_pbt_add() {
+    let mut rng = o1_utils::tests::make_test_rng(None);
+
+    let random_evaluation: [Fp; 6] = std::array::from_fn(|_| Fp::rand(&mut rng));
+    let p1 = unsafe { Dense::<Fp, 6, 4>::random(&mut rng) };
+    let p2 = unsafe { Dense::<Fp, 6, 4>::random(&mut rng) };
+    let p3 = p1.clone() + p2.clone();
+    let eval_p1 = p1.eval(&random_evaluation);
+    let eval_p2 = p2.eval(&random_evaluation);
+    let eval_p3 = p3.eval(&random_evaluation);
+    assert_eq!(eval_p3, eval_p1 + eval_p2);
+}
+
+#[test]
+fn test_eval_pbt_sub() {
+    let mut rng = o1_utils::tests::make_test_rng(None);
+
+    let random_evaluation: [Fp; 6] = std::array::from_fn(|_| Fp::rand(&mut rng));
+    let p1 = unsafe { Dense::<Fp, 6, 4>::random(&mut rng) };
+    let p2 = unsafe { Dense::<Fp, 6, 4>::random(&mut rng) };
+    let p3 = p1.clone() - p2.clone();
+    let eval_p1 = p1.eval(&random_evaluation);
+    let eval_p2 = p2.eval(&random_evaluation);
+    let eval_p3 = p3.eval(&random_evaluation);
+    assert_eq!(eval_p3, eval_p1 - eval_p2);
+}
+
+#[test]
+fn test_eval_pbt_mul_by_scalar() {
+    let mut rng = o1_utils::tests::make_test_rng(None);
+
+    let random_evaluation: [Fp; 6] = std::array::from_fn(|_| Fp::rand(&mut rng));
+    let p1 = unsafe { Dense::<Fp, 6, 4>::random(&mut rng) };
+    let c = Fp::rand(&mut rng);
+    let p2 = p1.clone() * Dense::<Fp, 6, 4>::from(c);
+    let eval_p1 = p1.eval(&random_evaluation);
+    let eval_p2 = p2.eval(&random_evaluation);
+    assert_eq!(eval_p2, eval_p1 * c);
+}
+
+#[test]
+fn test_eval_pbt_neg() {
+    let mut rng = o1_utils::tests::make_test_rng(None);
+
+    let random_evaluation: [Fp; 6] = std::array::from_fn(|_| Fp::rand(&mut rng));
+    let p1 = unsafe { Dense::<Fp, 6, 4>::random(&mut rng) };
+    let p2 = -p1.clone();
+    let eval_p1 = p1.eval(&random_evaluation);
+    let eval_p2 = p2.eval(&random_evaluation);
+    assert_eq!(eval_p2, -eval_p1);
+}