Today Senator Scott Brown is appearing at the White House to witness the signing of a bill he pushed in the Senate — part of Brown’s effort to build his credentials as bipartisan and independent of the national GOP.

There’s little doubt that Brown has successfully burnished those credentials, and he’s maintained high favorability ratings in the face of Elizabeth Warren’s challenge. A recent Boston Globe poll found that 49 percent see Brown as more willing than Warren to work across party lines, and 57 percent view him favorably.

But here’s the question: How much will Brown’s positive ratings matter to the outcome? The most decisive factor may end up being how Warren is viewed. The race may turn on whether Brown can drive up Warren’s negatives — not whether he can keep his own positives high.

The Globe poll finds that despite Brown’s clear success in coming across as bipartisan and likeable, the two are still statistically deadlocked at 37-35, with 26 percent undecided. Even though many Massachusetts voters have accepted Brown’s arguments about himself, one-fourth haven’t made up their minds.

What’s more, Dems believe the election will turn on roughly 800,000 voters who did not come out in the special election that Brown won in 2010 — but will come out this fall, because it’s a presidential year. Many of these voters are believed to be Obama supporters.

To win reelection, Brown will have to win over many voters who didn’t vote for him the first time — and who may be more receptive to the national Dem message than to the Republican one.

This is why job one for national Republicans and conservative groups has been to try to drive up Warren’s negatives. They’ve hammered her for months as a Harvard elitist and Occupy Wall Street sympathizer, in hopes of rendering her unacceptable to culturally conservative Dems and independents who will back Obama in November.

These voters may already see Brown as likeable and independent of the national GOP. Their vote, however, may turn on what they think of Warren. According to the Globe poll, Warren, too, remains well-liked — despite months of attacks on her character. Yes, Brown is more liked than Warren. But she is not disliked, either — and this could be key.

The pact against outside spending adds a twist: It’s on Brown and the national GOP to drive up her negatives — with a negative campaign.

None of this means Warren doesn’t face an extremely tough road ahead. Both sides think this will come down to the wire. But it may turn on this: If undecideds accept the Brown/GOP version of Warren — pointy-headed Harvard professor who thinks she knows how to spend ordinary folks’ money better than they do — he wins.

But if they accept her story as she tells it — maintenance man’s daughter who made good, went to Harvard, and emerged as a major fighter for the middle class, with the best ideas on how to fortify it — she wins.